# Number Theory

1. Three people each think of a number, which is a product of two different primes. The product of the three numbers which are thought of could be?

1. 120
2. 12100
3. 240
4. 3000
2. The GCD of ‘a’ and 72 is 24 and the LCM of ‘b’ and 24 is 72. Find the GCD of (a, b) and LCM of (a, b) given that a is the smallest three digit number having this property, and b is the biggest integer having this property.

b = 72
GCD(a , b) = 24
LCM(a , b) = 360
3. The sum and LCM of two positive integers x, y are given to be 40 and 48 respectively. Find the two integers.

Correct Answer: x = 24 & y = 16 or x = 16 & y = 24
4. Using all the digits 1 to 9 only once, how many nine digit prime numbers can you write?

1. 1
2. None
3. 9
4. More than 100
5. ‘a’ and ‘b’ are square numbers; the LCM of ‘a’ and 140 is 560 and LCM of b and 140 is 700. Then the LCM of a and b is?

1. 400
2. 1600
3. 2500
4. 4900
6. A three digit number with digits A, B, C in that order is divisible by 9. A is an odd digit and C is an even digit. B and C are non zero. The number of such three digit numbers is?

1. 4
2. 8
3. 16
4. 20
7. The number of non negative integers which are less than 1000 and end with only one zero is.

1. 90
2. 99
3. 91
4. 100
8. The digits of the year 2000 add up to 2. In how many years has it happened since the year 1 till this year (2021)?

1. 3
2. 6
3. 9
4. 10
9. A certain number has exactly eight factors including 1 and itself. Two of it’s factors are 21 and 35. The number is?

1. 105
2. 210
3. 420
4. 525
10. The least number of numbers to be deleted from the set {1,2,3,4,...... ,16} so that the product of the remaining numbers is a perfect square is?

1. 1
2. 2
3. 3
4. 4
11. The largest positive integer which cannot be written in the form 5m + 3n where m and n are positive integers is?

1. 30
2. 12
3. 7
4. 15
12. A = {a,b,c,d} is a set of four integers. We pick two integers out of A and add them. The following six sums are obtained - 0,2,4,8,10,12. Find the four integers in Set A?

Correct Answer: {a, b, c, d} = {-3, 3, 5, 7} or {-1, 1, 3, 9}
13. When 26 is divided by a positive integer N, the remainder is 2. The sum of all possible values of N is ________ .

14. If we multiply 2 by itself repeatedly four times we get 2 × 2 × 2 × 2 = 16 and it’s unit digit is 6. Suppose we multiply 2 with itself 2021 times we get a big number B. What is the unit digit of B?

15. Observe the sequence 9, 91, 19, 911, 191, 119, 9111, 1911, 1191, 1119, ...... What is the 45th term of the sequence?

16. The positive integers a, b satisfy 23a = 32b. Can a + b be a prime number? Justify your answer.

Correct Answer: a + b can't be a Prime number
17. Which of the following can never be a common factor of 287 + x and 378 + x where x can be any natural number?

1. 26
2. 13
3. 91
4. 7
26
18. A quiz has 20 questions with seven points awarded for each correct answer, two points deducted for each wrong answer and zero for each question omitted. Ram scores 87 points. How many questions did he omit?

1. 2
2. 5
3. 7
4. 9
5
19. How many four digit numbers with middle digit 97 are divisible by 45?

1. 0
2. 2
3. 4
4. 1
2
20. Show that it is impossible to find a positive integer such that the sum of its square and its cube is an integral multiple of the square of the next highest integer.

21. From the set 1,2,3, ...... , 2021, In how many ways can one choose two different numbers whose sum is an even number?

22. Let N be the greatest integral multiple of 8, such that no two of its digits are same. What is the remainder when N is divided by 1000.

23. How many times does the number 2 appear in a book having page numbers 1 to 250?

24. What is the greatest positive integer n which makes n3 + 100 divisible by n + 10?

25. The highest power of 2 that divides the sum of the numbers 4 + 44 + 444 + 4444 + 44444 + ...... + (100 4's) is

1. 2
2. 3
3. 4
4. 5