Number Theory

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  2. 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
    Correct Answer: Choice B
    Correct Answer

  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.

    Correct Answer: a = 120
    b = 72
    GCD(a , b) = 24
    LCM(a , b) = 360
    Correct Answer

  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
    Correct Answer

  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
    Correct Answer: Choice B
    Correct Answer

  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
    Correct Answer: Choice A
    Correct Answer

  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
    Correct Answer: Choice D
    Correct Answer

  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
    Correct Answer: Choice C
    Correct Answer

  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
    Correct Answer: Choice D
    Correct Answer

  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
    Correct Answer: Choice A
    Correct Answer

  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
    Correct Answer: Choice C
    Correct Answer

  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
    Correct Answer: Choice D
    Correct Answer

  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}
    Correct Answer

  13. When 26 is divided by a positive integer N, the remainder is 2. The sum of all possible values of N is ________ .

    Correct Answer: 57
    Correct Answer

  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?

    Correct Answer: 54 months
    Correct Answer

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

    Correct Answer: 111111119
    Correct Answer

  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
    Correct Answer

  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
    Correct Answer: Choice A
    26
    Correct Answer

  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
    Correct Answer: Choice B
    5
    Correct Answer

  19. How many four digit numbers with middle digit 97 are divisible by 45?

    1. 0
    2. 2
    3. 4
    4. 1
    Correct Answer: Choice B
    2
    Correct Answer

  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?

    Correct Answer: 1020100
    Correct Answer

  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.

    Correct Answer: 120
    Correct Answer

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

    Correct Answer: 106
    Correct Answer

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

    Correct Answer: 890
    Correct Answer

  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
    Correct Answer: Choice B
    3
    Correct Answer