CAT Questions | CAT Number Systems Questions

CAT Quantitative Aptitude | CAT Number Systems: Digits problems

A CAT Number theory question from Number Systems: Digits that appears in the Quantitative Aptitude section of the CAT Exam will consist of concepts from Digits, Test of Divisibility, Perfect squares and so on. In CAT Exam, one can expect to get 1~2 questions from CAT Number Systems: Digits. Make use of 2IIMs Free CAT Questions, provided with detailed solutions and Video explanations to obtain a wonderful CAT score. If you would like to take these questions as a Quiz, head on here to take these questions in a test format, absolutely free.

  1. CAT Number Theory: Product of Digits

    How many numbers with distinct digits are possible product of whose digits is 28?

    1. 6
    2. 4
    3. 8
    4. 12
    Choice C
    8

  2. CAT Number Theory: Sum of first n natural numbers

    A page is torn from a novel. The sum of the remaining page numbers is 10000. What is the sum of the two page-numbers on the torn page of this novel?

    1. 77 and 78
    2. 33 and 34
    3. 8 and 9
    4. None of these
    Choice D
    None of these

  3. CAT Number Theory: Perfect Square

    A 4-digit number of the form aabb is a perfect square. What is the value of a - b?

    1. 3
    2. 2
    3. 4
    4. 1
    Choice A
    3

  4. CAT Number Theory: Last two digits

    What are the last two digits of the number 745?

    1. 07
    2. 23
    3. 49
    4. 43
    Choice A
    07

  5. CAT Number Theory: Multiples

    n2 + 5n + 6 is a multiple of 6. n is natural number less than 100. How many values can n take?

    1. 33
    2. 65
    3. 66
    4. 67
    Choice C
    66

  6. CAT Number Theory: Greatest integer function

    [x] is the greatest integer less than or equal to x. Find the number of positive integers n such that [\\frac{n}{11}\\)] = [\\frac{n}{13}\\)] .

    1. 31
    2. 25
    3. 35
    4. 40
    Choice C
    35

  7. CAT Number Theory: Medians, Number

    Positive numbers 1 to 55, inclusive are placed in 5 groups of 11 numbers each. What is the maximum possible average of the medians of the 5 groups?

    1. 34
    2. 28
    3. 35
    4. 38
    Choice D
    38

  8. CAT Number Theory: 44th digit of a number

    N is an 80-digit positive integer (in the decimal scale). All digits except the 44th digit (from the left) are 2. If N is divisible by 13, find the 44th digit?

    1. 5
    2. 6
    3. 1
    4. 2
    Choice B
    6

  9. CAT Number Theory: Divisibility of powers

    A number 416 + 1 is divisible by x. Which among the following is also divisible by x?

    1. 496 + 1
    2. 432 + 1
    3. 48 + 1
    4. 448 + 1
    Choice D
    448 + 1

  10. CAT Number Theory: Test of Divisibility

    Which among the following is the smallest 7 digit number that is exactly divisible by 43?

    1. 1000043
    2. 1000008
    3. 1000006
    4. 1000002
    Choice B
    1000008

  11. CAT Number Theory: Three digit number combinations

    How many 3-digit positive integers, with digits a, b and c exist such that a < b and c < b? Assume that a is in hundred's place, b is in ten's place, c is in unit's place and a is a non-zero digit.

    1. 450
    2. 240
    3. 364
    4. 648
    Choice B
    240

The Questions that follow, are from actual CAT papers. If you wish to take them separately or plan to solve actual CAT papers at a later point in time, It would be a good idea to stop here.


  1. CAT 2020 Question Paper Slot 3 - Number Theory

    Let N, x and y be positive integers such that N = x + y, 2 < x < 10 and 14 < y < 23. If N > 25, then how many distinct values are possible for N?


  2. CAT 2020 Question Paper Slot 3 - Digits

    How many pairs (a,b) of positive integers are there such that a ≤ b and ab = 42017?

    1. 2019
    2. 2018
    3. 2020
    4. 2017
    Choice B
    2018

  3. CAT 2020 Question Paper Slot 2 - Digits

    If x and y are positive real numbers satisfying x + y = 102, then the minimum possible value of 2601(1 + \\frac{1}{x}))(1 + \\frac{1}{y})) is

    2704

  4. CAT 2020 Question Paper Slot 2 - Number Theory

    How many 4-digit numbers, each greater than 1000 and each having all four digits distinct, are there with 7 coming before 3?

    315

  5. CAT 2020 Question Paper Slot 1 - Number Theory

    How many 3-digit numbers are there, for which the product of their digits is more than 2 but less than 7?


  6. CAT 2020 Question Paper Slot 1 - Number Theory

    The mean of all 4 digit even natural numbers of the form 'aabb', where a>0, is

    1. 5544
    2. 4466
    3. 4864
    4. 5050
    Choice A
    5544

  7. CAT 2020 Question Paper Slot 1 - Digits

    Among 100 students, x1 have birthdays in January, x2 have birthdays in February, and so on. If x0 = max(x1, x2, ..., x12), then the smallest possible value of x0 is

    1. 8
    2. 10
    3. 12
    4. 9
    Choice D
    9

  8. CAT 2020 Question Paper Slot 1 - Digits

    If a, b and c are positive integers such that ab = 432, bc = 96 and c < 9, then the smallest possible value of a + b + c is

    1. 56
    2. 49
    3. 46
    4. 59
    Choice C
    46

  9. CAT 2019 Question Paper Slot 2 - Number theory

    Let a, b, x, y be real numbers such that a2 + b2 = 25 , x2 + y2 = 169 and ax + by = 65. If k = ay - bx, then

    1. k = 0
    2. k > \\frac{5}{13})
    3. k = \\frac{5}{13})
    4. 0 < k ≤ \\frac{5}{13})
    Choice A
    k = 0

  10. CAT 2019 Question Paper Slot 2 - Number theory

    How many pairs (m,n) of positive integers satisfy the equation the equation m2 + 105 = n2? [TITA]


  11. CAT 2019 Question Paper Slot 2 - Number theory

    In a six-digit number, the sixth, that is, the rightmost, digit is the sum of the first three digits, the fifth digit is the sum of first two digits, the third digit is equal to the first digit, the second digit is twice the first digit and the fourth digit is the sum of fifth and sixth digits. Then, the largest possible value of the fourth digit is [TITA]


  12. CAT 2019 Question Paper Slot 1 - Number theory

    The product of two positive numbers is 616. If the ratio of the difference of their cubes to the cube of their difference is 157 : 3, then the sum of the two numbers is

    1. 50
    2. 85
    3. 95
    4. 58
    Choice A
    50

  13. CAT 2018 Question Paper Slot 2 - Number Theory

    If the sum of squares of two numbers is 97, then which one of the following cannot be their product?

    1. 64
    2. -32
    3. 16
    4. 48
    Choice A
    64

  14. CAT 2018 Question Paper Slot 2 - Number Theory

    How many two-digit numbers, with a non-zero digit in the units place, are there which are more than thrice the number formed by interchanging the positions of its digits?

    1. 5
    2. 8
    3. 7
    4. 6
    Choice D
    6

  15. CAT 2018 Question Paper Slot 1 - Number Theory

    While multiplying three real numbers, Ashok took one of the numbers as 73 instead of 37. As a result, the product went up by 720. Then the minimum possible value of the sum of squares of the other two numbers is: [TITA]


  16. CAT 2018 Question Paper Slot 1 - Number theory

    How many numbers with two or more digits can be formed with the digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 so that in every such number, each digit is used at most once and the digits appear in the ascending order?[TITA]

      502

    1. CAT 2017 Question Paper Slot 2 - Number Theory

      The numbers 1, 2,..., 9 are arranged in a 3 X 3 square grid in such a way that each number occurs once and the entries along each column, each row, and each of the two diagonals add up to the same value. If the top left and the top right entries of the grid are 6 and 2, respectively, then the bottom middle entry is: [TITA]


    2. CAT 2017 Question Paper Slot 2 - Number Theory

      If the product of three consecutive positive integers is 15600 then the sum of the squares of these integers is

      1. 1777
      2. 1785
      3. 1875
      4. 1877
      Choice D
      1877

    3. CAT 2017 Question Paper Slot 2 - Number Theory

      How many different pairs (a, b) of positive integers are there such that a ≤ b and \\frac{1}{a}) + \\frac{1}{b}) = \\frac{1}{9}) ? [TITA]


    The Questions that follow, are from actual XAT papers. If you wish to take them separately or plan to solve actual XAT papers at a later point in time, It would be a good idea to stop here.


    1. XAT 2020 Question Paper - QADI

      When expressed in a decimal form, which of the following numbers will be nonterminating as well as non-repeating?

      1. ( 𝜋/2)[(1/𝜋)+1] - 𝜋/2
      2. sin21° + sin22° + … + sin289°
      3. ∛(729 )/3 + 22/7
      4. (4 - 𝜋)[1 + ( 𝜋/4 ) + ( 𝜋/4 )2 + ( 𝜋/4 )3 + ….(infinite terms)]
      5. √𝟐( 𝟑√𝟐 - 4 √𝟐) + √𝟑
      Choice E√𝟐( 𝟑√𝟐 - 4 √𝟐) + √𝟑

    2. XAT 2020 Question Paper - QADI

      A shop sells bags in three sizes: small, medium and large. A large bag costs Rs.1000, a medium bag costs Rs.200, and a small bag costs Rs.50. Three buyers, Ashish, Banti and Chintu, independently buy some numbers of these types of bags. The respective amounts spent by Ashish, Banti and Chintu are equal. Put together, the shop sells 1 large bag, 15 small bags and some medium bags to these three buyers. What is the minimum number of medium bags that the shop sells to them?

      1. 7
      2. 5
      3. 9
      4. 4
      5. 10
      Choice A
      7

    3. XAT 2020 Question Paper - QADI

      Consider the four variables A, B, C and D and a function Z of these variables,Z = 15A2 − 3B4 + C + 0.5D. It is given that A, B, C and D must be non-negative integers and that all of the following relationships must hold:
      i) 2A + B ≤ 2
      ii) 4A + 2B + C ≤ 12
      iii) 3A + 4B + D ≤ 15 If Z needs to be maximised, then what value must D take?

      1. 15
      2. 12
      3. 0
      4. 10
      5. 5
      Choice B
      12

    4. XAT 2019 Question Paper - QADI

      Given that a and b are integers and that 5 + 2√7 is a root of the polynomial x2 – ax + b + 2√7 in x, what is the value of b?

      1. 4
      2. 2
      3. 9
      4. 7
      5. 14
      Choice B
      2

    5. XAT 2019 Question Paper - QADI

      We have two unknown positive integers m and n, whose product is less than 100.
      There are two additional statement of facts available:
      1. mn is divisible by six consecutive integers { j, j + 1,...,j + 5 }
      2. m + n is a perfect square.

      Which of the two statements above, alone or in combination shall be sufficient to determine the numbers m and n?

      1. Each statement alone is sufficient to answer the question.
      2. Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.
      3. Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.
      4. Statements 1 and 2 together are not sufficient, and additional data is needed to answer the question.
      5. Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.
      Choice B
      Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

    6. XAT 2018 Question Paper - QADI

      Find the value of the expression: 10 + 103 + 106 + 109

      1. 1010101010
      2. 1001000010
      3. 1001000110
      4. 1001001010
      5. 100010001010
      Choice D
      1001001010

    7. XAT 2018 Question Paper - QADI

      X and Y are the digits at the unit's place of the numbers (408X) and (789Y) where X ≠ Y. However, the digits at the unit's place of the numbers (408X)63 and (789Y)85 are the same. What will be the possible value(s) of (X + Y)?
      Example: If M = 3 then the digit at unit's place of the number (2M) is 3 (as the number is 23) and the digit at unit's place of the number (2M)2 is 9 (as 232 is 529).

      1. 9
      2. 10
      3. 11
      4. 12
      5. None of these
      Choice B
      10

    8. XAT 2018 Question Paper - QADI

      This question is followed by two statements. These statements provide data that mayhelp answer the respective questions. Read the questions and the statements and determine if the data provided by the statements is sufficient or insufficient, on their own or together, to answer the questions.

      Accordingly, choose the appropriate option given below the questions.
      Harry and Sunny have randomly picked 5 cards each from a pack of 10 cards, numbered from 1 to 10. Who has randomly picked the card with number 2 written on it?
      Statement I: Sum of the numbers on the cards picked by Harry is 5 more than that of Sunny.
      Statement II: One has exactly four even numbered cards while the other has exactly four odd numbered cards.

      1. Statement I alone is sufficient to answer.
      2. Statement II alone is sufficient to answer.
      3. Either of the statement is sufficient to answer.
      4. Both statements are required to answer.
      5. Additional information is required.
      Choice D
      Both statements are required to answer.

    The Questions that follow, are from actual IPMAT papers. If you wish to take them separately or plan to solve actual IPMAT papers at a later point in time, It would be a good idea to stop here.


    1. IPMAT 2020 Sample Paper - IPM Rohtak Quants

      If * = +, / = -, + = *, - = / then 43 * 561 + 500 - 100 / 10 = ?

      1. 3211
      2. 2388
      3. 2883
      4. 2838
      Choice D
      2838

    2. IPMAT 2020 Sample Paper - IPM Rohtak Quants

      John's present age is one fourth of his father's age two years ago. John's father's age will be twice Raman's age after 10 years. If Raman's 12th birthday was celebrated 2 years ago, then what is John's present age?

      1. 5 years
      2. 7 years
      3. 9 years
      4. 11 years
      Choice C
      9 years

    3. IPMAT 2020 Sample Paper - IPM Rohtak Quants

      Simplification: \25^{(2.7)} \times 5^{(4.2)} \div 5^{(5.4)} = ?\\)

      1. 54
      2. 5(3.2)
      3. 5(4.1)
      4. 5(4.2)
      Choice D
      5(4.2)

    4. IPMAT 2020 Sample Paper - IPM Rohtak Quants

      18800 / 470 / 20

      1. 1
      2. 2
      3. 3
      4. 4
      Choice B
      2

    5. IPMAT 2020 Question Paper - IPM Indore Quants

      In a four-digit number, the product of thousands digit and units digit is zero while their difference is 7. Product of the middle digits is 18 . The thousands digit is as much more than the units digit as the hundreds digit is more than the tens digit. The four-digit number is __________.

      7920

    6. IPMAT 2019 Question Paper - IPM Indore Quants

      Assume that all positive integers are written down consecutively from left to right as in 1234567891011...... The 6389th digit in this sequence is


    7. IPMAT 2019 Question Paper - IPM Indore Quants

      The number of pairs of integers whose sums are equal to their products is


    8. IPMAT 2019 Question Paper - IPM Indore Quants

      How many different numbers can be formed by using only the digits 1 and 3 which are smaller than 3000000 ?

      1. 64
      2. 128
      3. 190
      4. 254
      Choice C
      190

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