2. Total Gadha's Complete Book of. NUMBER SYSTEM If n is an odd natural number, what is the highest number that always divides n × (n2 – 1)?. Answer: n . Totalgadha Number System Book - Download as PDF File .pdf), Text File .txt) or read online. Good book on number system. View Complete Book of Number System (Total Gadha) from BSC at University of South Africa. 1 http:/lycgodoomcari.gq Total Gadhas Complete Book of.

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lycgodoomcari.gq . In a number system the product of 44 and 11 is system, when converted to the decimal number system, becomes. 1. . definitely the 3 e books by total gadha number systems, is view test prep - complete book of geometry - total gadha ().pdf from maths. Total lycgodoomcari.gq Free Download Here The Complete Book of Number System - MBAtutes.

We will look at different books for CAT which students should refer to in order to perk up their study. Quantitative Aptitude is one section for which the coaching institutes have developed the most comprehensive preparation material. There are loads of books in the market which can be bought by students in this section. There are a lot of students who are in fact searching for pdf of Arun Sharma online. Check the book over here. SK Verma — If you want to improve your basics in quant, then this is a great option. The book goes into much detail and may get a tad boring for those who are used to this section. Those who are brand new to quant, should download this book over Arun Sharma. Details over here. TIME material of Quant is pretty good. Total Gadha Number System Book. Arun Sharma, complete it twice again. It is probably one of the most important books for CAT.

What is the difference in their payments? The annual sales of a company in yr was rs and in the yr was rs A test consists of 4 sections each of 45 marks as their maximum.

Find the number of ways in which one can score 90 or more. There are 5 botles of sherry and each has its own cap. How many ways are there so that not a single cap is not on the correct bottle.

A number when divided by leaves a quotient Q and a remainder R. None of these How many values of k are possible? For how many values of n, Q is a perfect square?

More than 4 Disclaimer- The following problems are from TotalGadha. How many three-digit numbers are there such that no two adjacent digits of the number are consecutive? All win went to the market and bought some chikoos, mangoes, and bananas.

Allwin bought 42 fruits in all. The number of bananas is less than half the number of chikoos; the number of mangoes is more than one-third the number of chikoos and the number of mangoes is less than three-fourths the number of bananas. In an organisation there are 40 employees belonging to different departments A, B and C.

Therefore, we first find the remainders when this number is divided by 9 and 4. The remainder by 9 would be the remainder when the sum of digits is divided by 9. Therefore, to find the remainder we need to find the smallest multiple of 4 that gives remainder 1 with 9. Find the remainder when … is divided by The overall remainder would be the smallest number that gives remainder 3 with 9 and remainder 2 with 4.

A number when divided by 8 leaves remainder 3 and quotient Q. What is the number? Let the divisor be D and the remainder be R. The HCF of 50 and is Therefore, the highest number can be What is the remainder when It can be proved that a number formed by writing any single digit 3 n times will be divisible by 3 n. This is left to students to check it out. Now, 33 is divisible by 11 but not If both are divisible by 11, their product is divisible by but 33 is divisible only by 11 therefore the expression is not divisible by If both are not divisible by 11, the expression is again not divisible by The number 24 is divisible by 1, 2, 3, 4, 6, 8, 12, and Hence all these numbers are divisors of How to find the number of divisors of a number: Let us the find the number of divisors of Any divisors of 60 will have powers of 2 equal to either 2 0 or 2 1 or 2 2.

Similarly, any divisor of 60 will have powers of 3 equal to either 3 0 or 3 1 , and powers of 5 equal to either 5 0 or 5 1. To make a divisor of 60, we will have to choose a power of 2, a power of 3 and a power of 5. Similarly, a power of 3 can be chosen in 2 ways and a power of 5 can be chosen in 2 ways.

Notice that we have added 1 each to powers of 2, 3 and 5 and multiplied. Now for the formula: Find the number of divisors of How many divisors of are odd numbers? An odd number does not have a factor of 2 in it. Therefore, we will consider all the divisors having powers of 3 and 5 but not 2. How many divisors of are even numbers?

How many divisors of are not divisors of and how many divisors of are not divisors of ? The best option here is to find the number of common divisors of and For that we find the highest common powers of all the common prime factors in and Therefore, the two numbers will have 18 factors in common. For unit digit equal to 5, the number has to be a multiple of 5 and it should not be a multiple of 2 otherwise the unit digit will be 0. How many divisors of 36 36 are perfect cubes?

To find the divisors which are perfect cubes, we need to take those powers of prime factors which are multiples of 3.

Both are 25 in number. Reverse Operations on Divisors: Find all the numbers less than which have exactly 8 divisors.

To find the number of divisors of a number, we used to add 1 to powers of all the prime factors and then multiply them together. Now, given the number of divisors, we will express this number as a product and then subtract 1 from every multiplicand to obtain the powers.

Therefore, the number is of the form a 1 b 1 c 1 , where a, b and c are prime. Therefore, the number is of the form a 3 b, where a and b are prime.

The number can also be of the form a 7 , but there is no such number less than Find the smallest number with 15 divisors. To find the smallest such number, we give the highest power to smallest rime factor, i. In a perfect square, all the prime factors have even powers.

We can select an even power of 2 in 3 ways, even power of 3 in 2 ways, and even power of 5 in 2 ways. All the even divisors of the number will have powers of 2 equal to one of 2, 2 2 , 2 3 , 2 4 , or 2 5.

What is the product of the elements of AUB? AUB will have all the divisors of and with the common divisors written only once.

Therefore, these common divisors will be multiplied only once. Therefore, divisors occur in pairs for numbers which are not perfect squares. The divisors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and Therefore, divisors occur in pairs except for the square root for numbers which are perfect squares. Also, since we are asked for integers, the pair consisting of two negative integers will also suffice.

We have to assign these prime factors and their powers to one of the two factors. As the two factors will be prime to each other, we will have to assign a prime factor with its power for example completely to one of the factors. For every prime factor, we have two ways of assigning it. Number of numbers less than or prime to a given number: The above formula is extremely versatile as it lets us find not only the numbers which do not contain any of the prime factors of N but also the numbers which do not contain some selected prime factors of N.

The following examples will make it clear: How many of the first natural numbers are not divisible by any of 2, 3 and 5? Therefore, we need to find the number of numbers which are less than and prime to Unlike the previous problem, this problem only asks for number not divisible by only 2 factors of , i. Therefore, in the formula we remove the part containing the factor of 3 and calculate the numbers of numbers prime to with respect to prime factors 2 and 5. These are in numbers in all we have already calculated it.

Which natural number has the highest number of divisors? Now we are looking for highest multiple of that is less than To calculate units digit of we only consider the units digit of Hence, we find the units digit of 7 Hence, we find the units digit of 3 x 4, respectively.

To calculate units digit of x y where x is a single digit number To calculate units digit of numbers in the form x y such 7 , 8 93 , 3 74 etc. When y is NOT a multiple of 4 We find the remainder when y is divided by 4.

The units digit of x y is the units digit of x r. When y is a multiple of 4 We observe the following conditions: Even numbers 2, 4, 6, 8 when raised to powers which are multiple of 4 give the units digit as 6. Odd numbers 3, 7, and 9 when raised to powers which are multiple of 4 give the units digit as 1.

Find the units digit of 7 The remainder when 33 is divided by 4 is 1. The units digit of 43 47 can be found by finding the units digit of 3 We have to find the units digit of 8 28 — 4 Since 28 and 24 are both multiples of 4, the units digits of both 8 28 and 4 24 will be 6. Hence the units digit of the difference will be 0.

Find the units digit of 43 43 — 22 Units digit of 43 43 is 7 and units digit of 22 22 is 4. Find the units digit of 3 3 3 Answer: Again, we find the remainder when the power is divided by 4.

Therefore, we find the remainder when 3 3 is divided by 4. Therefore, we find the remainder when 17 13 11 is divided by 4. The sum of these units digits gives a unit digit of 5. Now these units digit will repeat 10 times each.

What are the last two digits of 31 ? Now, the second term will end with one zero and the tens digit of the second term will be the product of and 3 i. Therefore, the last two digits of the second term will be The last digit of the first term is 1. So the last two digits of 31 are Now, here is the shortcut: Multiply the tens digit of the number 3 here with the last digit of the exponent 6 here to get the tens digit. The units digit is equal to one. Here are some more examples: Once the number is ending in 1 we can straight away get the last two digits with the help of the previous method.

Convert the number till the number gives 1 as the last digit and then find the last two digits according to the previous method. Now try the method with a number ending in 7: Find the last two digits of 87 Find the last two digits of: Therefore, our purpose is to get 76 as last two digits for even numbers.

We know that 24 2 ends in 76 and 2 10 ends in Also, 24 raised to an even power always ends with 76 and 24 raised to an odd power always ends with Therefore, 24 34 will end in 76 and 24 53 will end in Find the last two digits of 2 Here if you need to multiply 76 with 2 n , then you can straightaway write the last two digits of 2 n because when 76 is multiplied with 2 n the last two digits remain the same as the last two digits of 2 n.

Here is an example: Find the last two digits of 64 We can find the last two digits of both the parts separately. Here are some examples: Find the last two digits of 62 Find the last two digits of 56 Find the highest power of 2 in 50! The highest power of 2 in 50! Now 5 is the largest prime factor of 30, therefore, the powers of 5 in 50! Therefore, there cannot be more 30s than there are 5 in 50! So we find the highest power of 5 in 50! The highest power of 5 in 50! Hence the highest power of 30 in 50!

We get a zero at the end of a number when we multiply that number by So, to calculate the number of zeroes at the end of ! The highest power of 5 in !

To find the number of divisors of 15! The prime factors in 15! Are , 3, 5, 7, 11 and Powers of 2 in 15! What is the rightmost non-zero digit in 15!? We saw that 15! Highest power of prime number p a in n! Find the highest power of 72 in ! Therefore, we need to find the highest power of 8 and 9 in 72!. Therefore, we need to find the highest power of 8 and 3 in !

Find the number of possible values of n if n is a three digit number. We can see that increasing the natural number by 1, we are gathering 3 more powers of 5. A number is divisible by 2, 4, 8, 16, 32,.. The number is divisible by 4 because the number formed by the last two digits, 64 is divisible by 4.

A digit number is formed by writing first 55 natural numbers next to each other. Find the remainder when the number is divided by Divisibility by 3 and 9 A number is divisible by 3 or 9 when the sum of the digits of the number is divisible by 3 or 9 respectively. The six-digit number 73A is divisible by 6. How many values of A are possible? Since the number is ending in an even digit, the number is divisible by 2.

To find divisibility by 3, we need to consider sum of the digits of the number. For the number to be divisible by 3, the sum of the digits should be divisible by 3. Hence A can take values equal to 0, 3, 6, and 9. Whenever we have to check the divisibility of a number N by a composite number C, the number N should be divisible by all the prime factors the highest power of every prime factor present in C. Divisibility by 7, 11, and 13 Let there be a 6- digit number abcdef. Starting from right to left, make groups of three digit numbers successively and continue till the end.

It is not necessary that the leftmost group has three digits. Grouping of the above number in groups of three, from right to left, is done in the following manner kj,ihg,fed,cba Add the alternate groups 1 st , 3 rd , 5 th etc..

If D is divisible by 7, then the original number is divisible by 7. If D is divisible by 11, then the original number is divisible by 11 If D is divisible by 13 then the original number is divisible by Any six-digit, or twelve-digit, or eighteen-digit, or any such number with number of digits equal to multiple of 6, is divisible by EACH of 7, 11 and 13 if all of its digits are same.

For example , etc. Find if the number is divisible by 7. We make the groups of three as said above- 29,, We can see that D is divisible by 7.

Hence, the original number is divisible by 7. Find the digit A if the number …A… is divisible by 7, where both the digits 8 and 9 are 50 in number. We know that and will be divisible by 7. Hence 8 written 48 times in a row and 9 written 48 times in a row will be divisible by 7. Hence we need to find the value of A for which the number 88A99 is divisible by 7.

Find a four-digit number abcd with distinct digits which is divisible by 4, such that bacd is divisible by 7, acbd is divisible by 5, and abdc is divisible by 9. Now, sum of the digits should be equal to 9, 18 or 27 as the number abdc is divisible by 9. Case 3: Case 4: A number consisting entirely of the digit one is called a repunit; for example, Find the smallest repunit that is divisible by A number formed by repeating a single digit 6 times is divisible by 7.

Also, the sum of digits is divisible by 9. The product of a two-digit number by a number consisting of the same digits written in the reverse order is equal to Find the lower number? As the R. Therefore, the number is The number abc is divisible by 7, 8 and 9.

The LCM of 7, 8, and 9 is Therefore, abc should be divisible by Therefore, 19abc should be divisible by The buckets that are available to you all have sizes that are powers of 3, i.

Which buckets do you use to fill the tank in the minimum possible time? You will certainly tell me that the first bucket you will use is of L. That will leave L of the tank still empty. The next few buckets you will use will L, 27 L and 1 L. The use of buckets can be shown as below http: The number 1 has been written in increasing powers of 3. The number 10 is called the 'base' in which this number was written.

Let a number abcde be written in base p, where a, b, c, d and e are single digits less than p. Conversion from any base to base ten The number pqrstu b is converted to base 10 by finding the value of the number.

Convert 5 to base The remainders, written in reverse order, give the equivalent number in base 'b'. Write the number 25 in base 4. Writing the remainders in reverse order the number 25 in base 10 is the number in base 4. Addition, subtraction and multiplication in bases: Add the numbers 7 and 7 Answers: The numbers are written as http: The Quotient is 1 and written is 2.

The Remainder is placed at the units place of the answer and the Quotient gets carried over to the ten's place. We obtain At the tens place: Subtract from in base 7. In the units column since 1 is smaller than 6, we carry the value equal to the base from the number on the left. Since the base is 7 we carry 7. Hence we write 2 in the units column. We proceed the same way in the rest of the columns. Important rules about bases A number in base N is divisible by N — 1 when the sum of the digits of the number in base N is divisible by N — 1.

The number 35A is in base 9. This number is divisible by 8. Find the value of digit A. The number will be divisible by 8 when the sum of the digits is divisible by 8. A four-digit number N 1 is written in base A new four-digit number N 2 is formed by rearranging the digits of N 1 in any order.

Then the difference N 1 — N 2 is divisible by a 9 b 10 c 12 d 13 Answer: Let the base be b. A positive whole number M less than is represented in base 2 notation, base 3 notation, and base 5 notation.

It is found that in all three cases the last digit is 1, while in exactly two out of the three cases the leading digit is 1. Whenever we change a number from base 10 to any other base, the units digit is the first remainder when the number is divided by that base. Therefore, M when divided by 2, 3 and 5 gives remainder 1 in each case.

Out of these 3 numbers, only the number 91 satisfies the second criterion of leading digit last remainder. A palindromic number reads the same forward and backward. A digit palindromic number in base 16 will always be divisible by Answer: Therefore, when written in base 5, the first remainders will be 0, 1 or 2. Working in base 7: This chapter will answer all these questions. There are two methods to find HCF of the given numbers: Prime Factorization Method- When a number is written as the product of prime numbers, the factorization is called the prime factorization of that number.

For every prime factor common to all the numbers, we choose the least index of that prime factor among the given number. The HCF is product of all such prime factors with their respective least indices. Find the HCF of 72, , and Answer: The lowest indices of 2 and 3 in the given numbers are 3 and 2 respectively.

Find the HCF of 36x 3 y 2 and 24x 4 y. The least index of 2, 3, x and y in the numbers are 2, 1, 3 and 1 respectively. Division method- To find HCF of two numbers by division method, we divide the higher number by the lower number.

Then we divide the lower number by the first remainder, the first remainder by the second remainder The last divisor is the required HCF. Find the HCF of and by the division method. Hence, the last divisor 72 is the HCF of and Three company of soldiers containing , , and soldiers are to be broken down into smaller groups such that each group contains soldiers from one company only and all the groups have equal number of soldiers.

What is the least number of total groups formed? The least number of groups will be formed when each group has number of soldiers equal to the HCF. The HCF of , and is Therefore, the numbers of groups formed for the three http: The numbers , and when divided by a number N give the same remainder of Find the highest such number N.

Since all the numbers give a remainder of 12 when divided by N, hence — 12 , — 12 and — 12 are all divisible by N. Hence, N is the HCF of , and The numbers , and , when divided by a number N, give the remainders of 22, 23 and 24 respectively. Find the greatest such number N. Hence, N will be the HCF of , and The HCF of two numbers is 12 and their sum is How many pairs of such numbers are possible? If the HCF if 12, the numbers can be written as 12x and 12y, where x and y are co-prime to each other.

The pair of numbers that are co-prime to each other and sum up to 24 are 1, 23 , 5, 19 , 7, 17 and 11, Hence, only four pairs of such numbers are possible. The numbers are 12, , 60, , 84, and , The HCF of two numbers is 12 and their product is How many such numbers are possible? Let the numbers be 12x and 12y, where x and y are co-prime to each other. Now we need to find co-prime pairs whose product is Therefore, the co-prime pairs will be 1, and 8, Therefore, only two such numbers are possible.

Find the HCF of 2 — 1 and 2 — 1 Answer: To calculate the LCM of two or more numbers, we use the following two methods: Prime Factorization Method: After performing the prime factorization of the numbers, i.

The LCM is the product of all these prime numbers with their respective highest indices. Find the LCM of 72, and The prime numbers present are 2, 3 and 5. The highest indices powers of 2, 3 and 5 are 5, 3 and 1, respectively. Find the LCM of 36x 3 y 2 and 24x 4 y. The highest indices of 2, 3, x and y are 3, 2, 4 and 2 respectively. Division Method: To find the LCM of 72, and , we use the division method in the following way: Note- This formula is applicable only for two numbers.

Hence both the sum and differences of the two numbers are divisible by the HCF. Find the highest four-digit number that is divisible by each of the numbers 24, 36, 45 and The highest four-digit number is Find the highest number less than that is divisible by each of the numbers 2, 3, 4, 5, 6 and 7. The LCM of 2, 3, 4, 5, 6 and 7 is Hence , and every multiple of , is divisible by each of these numbers. Hence, the number , , , and are all divisible by each of these numbers. We can see that is the highest number less than which is multiple of Hence, the highest number divisible by each one of 2, 3, 4, 5, 6 and 7, and less than is Find the lowest number which gives a remainder of 5 when divided by any of the numbers 6, 7, and 8.

The LCM of 6, 7 and 8 is Hence, is divisible by 6, 7 and 8. What is the smallest number which when divided by 9, 18, 24 leaves a remainder of 5, 14 and 20 respectively? Now the LCM of 9, 18, and 24 is Therefore, if we subtract 4 from 72, the resulting number will give remainders of 5, 14, and 20 with 9, 18, and A number when divided by 3, 4, 5, and 6 always leaves a remainder of 2, but leaves no remainder when divided by 7.

What is the lowest such number possible? We can see that is divisible by 7. For how many pairs a, b of natural numbers is the LCM of a and b is 2 3 5 7 11 13? Let's solve for the powers of 2. One of the number will have 2 3 in it, as the LCM has 2 3. Now the other number can have the powers of 2 as 2 0 , 2 1 , 2 2 , and 2 3.

Therefore, number of pairs will be 4: All the digits of N 2 are added to obtain a number N 3 , and so on, till we obtain a single digit number N. This single digit number N is called the digit sum of the original number N 1. Hence, the digit sum of the number is 9. In finding the digit-Sum of a number we can ignore the digit 9 or the digits that add up to 9. For example, in finding the digit-sum of the number , we can ignore the digits 2, 6, 1, and 9.

Digit-Sum Rule of Multiplication: The digit-sum of the product of two numbers is equal to the digit sum of the product of the digit sums of the two numbers! The product of and 35 is A quick check will show that the digit-sum of the product is 3. The digit-sums of the individual numbers , and are 1, 9, and 5. Hence, the answer obtained by multiplication is not correct. Although the answer of multiplication will not be correct if the digit-sum of the product of the digit-sums is not equal to digit-sum of the product, but the reverse is not true i.

What is that number? Digit sums on both sides will be the same. If one of the multiplicand is 9, the digit sum is always 9. As one of the multiplicands is 9, the digit sum will be 9. Determining if a number is a perfect square or not S. Numbe r Squar e Digit- Sum of the S. Numbe r Squar e Digit-Sum of the square http: A number will NOT be a perfect square if its digit-sum is NOT 1, 4, 7, or 9, but it may or may not be a perfect square if its digit-sum is 1, 4, 7, or 9.

Is the number a perfect square? Hence, the number cannot be a perfect square. Is N be a perfect square? We can see that the digit sum of a perfect square is always 1, 4, 7, or 9. As the digit sum of the number is 3, it cannot be a perfect square. If a five-digit number N is such that the sum of the digits is 29, can N be the square of an integer? To put it simply, if the given number is an integer, then the greatest integer gives the number itself, otherwise it gives the first integer towards the left of the number of x on the number line.

For example, [1. We can see that [1. Note that the red dot indicates that integer value on the number line is not included while the green dot indicates that the integer value is included. The product of three consecutive odd numbers is What is the sum of the three numbers? Then, the number of divisors of N is A. In the value of the number 30! Then, the unit digit of the number that is left is A. How many different values of N are possible? In , my age was equal to the last two digits of my birth year.

My grandfather said that it was true for him also. N is the sum of the squares of three consecutive odd numbers such that all the digits of N are the same. If N is a four-digit number, then the value of N is http: While driving on a straight road, Jason passed a milestone with a two-digit number. After exactly an hour, he passed a second milestone with the same two digits written in reverse order. Exactly one more hour after that, he passed a third milestone with the same two digits reversed and separated by a zero.

What is the sum of the two digits? The squares of the natural numbers are written in a straight line … to form a digits number. What is the th digit from the left? Then the sum of the digits of S is A. Swadesh threw five standard dice simultaneously. He found that the product of the numbers on the top faces was Which of the following could not be the sum of the numbers on the top five faces?

The last two digits of S are A. How many natural numbers between 1 and are NOT multiples of any of the numbers 2, 3, or 5?

The last two digits of 4 are A. The numbers and are multiplied. All the divisors of 72 are multiplied. The number AB, where A and B are single-digit numbers, is divisible by N is the smallest natural number which when multiplied by 7 gives a product P.

Every digit of P is one. The product when N is multiplied by 8 is A. The remainder when 7 77 is divided by 9 is A. In the nineteenth century a person was X years old in the year X 2.

How old was he in ? The average of the nine numbers 9, 99, ,…, is A. Then A does not contain the digit A. How many natural numbers less than 65 have odd number of divisors including 1 and those numbers themselves? The person who says 'five' is taken out of the line. Those remaining repeat this procedure until only four people remain in the line. What was the original position in the line of the last person to leave? What is the sum of the sum of the sum of the digits of 55!?

An Indian king was born in a year that was a square number, lived a square number of years and died in a year that was also a square number. Then the year he could have been born in was A.

To number the pages of a book, exactly digits were used. How many pages did the book have? The highest power of 12 that can divide 5 36 — 1 is A. What is the largest prime whose cube divides 1! The number 2n! I and II only B. I and III only D. Baghira, the oldest inmate of Tihar Jail is learning mathematics. He notices the following facts about his prisoner number: It is a three digit number not bigger than If you sum the cube of the digits of the number, you get the number itself.

The number is the sum of consecutive factorials. The number in base 10 is written as b. Then, the base b is equal to A. His Mercedes registration number is a four-digit perfect square formed by repeating the rightmost digit of his nightclub number. The teacher of Confusius, the confused soul, told him: If you reverse the digits of my age and subtract the number from my age you again have a perfect square.

The remainder when 2 is divided by 13 is A. How many positive integers less than or equal to are relatively prime to ? The difference between the cubes of two consecutive positive integers is Then the product of these integers is A. How many integers between 1 and , both inclusive, can be expressed as the difference of the squares of two non negative integers? The product P of three positive integers is 9 times their sum, and one of the integers is the sum of the other two.

The sum of all possible values of P is A. Let M be the greatest number divisible by 8, such that no digit from 0 to 9 is repeated in M. What is the remainder when M is divided by ? Let q and r be the quotient and remainder when M, a five digit number, is divided by Then I. A, B and C can be equal for some value of x. A, B and C can all take different values for some value of x.

I is true but II is false B. II is true but I is false C. Both I and II are true D. Both I and II are false How many perfect squares are the divisors of the product 1! The number ! How many zeroes are there at the end? The unit digit of 7 7 7 is A. In how many ways can the number be written as a sum of two or more consecutive positive integers? It is given that m and n are two smallest natural numbers satisfying following conditions I.

Five consecutive integers are chosen. Let S denote their sum and let P denote their product. If the product of four positive integers is 10! What is the smallest possible value their sum can have? S is a six digit number beginning with 1. If the digit 1 is moved from the leftmost place to the rightmost place the number obtained is three times of S. Which of the following numbers can be written as the sum of the squares of three odd natural numbers?

In a bag, some slips of paper are kept with the numbers thirteen or fourteen written on them. The slips with number thirteen written on them are five more than the slips with number fourteen written on them. Which of the following can be the sum of the numbers in the bag?

Two natural numbers a and b are given in base How many two-digit positive integers are there which are one and a half times larger than the product of their digits? A three-digit number abc is divisible by 7 if A. For how many values of k is 12 12 the least common multiple of 6 6 , 8 8 , and k? All the divisors of , including 1 and the number itself, are summed up. The sum is What is the sum of the reciprocals of all the divisors of ?

Then the set of intersection of all values of S and T is A. What is the remainder when the number digits Yes B. Maybe D. Vinay has boxes with him. He has to put least oranges in one box and oranges at the most. Then the least number of boxes containing the same number of oranges is A. There were 90 questions in an exam. In a class, the teacher wrote a set of consecutive integers beginning with 1 on the blackboard.

Little Johnny came and erased one number. The average of the remaining numbers was 13 4 What was the number that little Johnny erased? What is the remainder when digits digits 10 If 11 sweets are distributed among four boys, then, which of the following is true? Two boys each received more than 1 sweet B. One of the boys received more than 3 sweets http: One of the boys received fewer than 3 sweets D.

One of the boys received exactly 2 sweets All possible pairs are formed from the divisors of How many such pairs have HCF of 45? How many natural numbers between 1 and have exactly four factors?