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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.

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
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
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
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
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
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
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
CAT Number Theory: 44^th digit of a number

N is an 80-digit positive integer (in the decimal scale). All digits except the 44^th digit (from the left) are 2. If N is divisible by 13, find the 44^th digit?
1. 5
2. 6
3. 1
4. 2
Choice B
6
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
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
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.

CAT 2022 Slot 2 - QA
If \(a\) and \(b\) are non-negative real numbers such that \(a+2 b=6\), then the average of the maximum and minimum possible values of \((a+b)\) is
1. 4
2. 4.5
3. 3.5
4. 3
Choice B
4.5
CAT 2022 Slot 1 - QA
Let \(a\) and \(b\) be natural numbers. If \(a^2+a b+a=14\) and \(b^2+a b+b=28\), then \((2 a+b)\) equals
Choice D
8
CAT 2021 Slot 3 - QA
A four-digit number is formed by using only the digits 1, 2 and 3 such that both 2 and 3 appear at least once. The number of all such four-digit numbers is
50
Correct: 4.28%
Incorrect: 54.08%
Unattempted: 41.64%
CAT 2021 Slot 2 - QA
For a 4-digit number, the sum of its digits in the thousands, hundreds and tens places is 14, the sum of its digits in the hundreds, tens and units places is 15, and the tens place digit is 4 more than the units place digit. Then the highest possible 4-digit number satisfying the above conditions is
4195
Correct: 22.54%
Incorrect: 28.07%
Unattempted: 49.39%
CAT 2021 Slot 1 - QA
The natural numbers are divided into groups as (1), (2, 3, 4), (5, 6, 7, 8, 9), ….. and so on. Then, the sum of the numbers in the 15th group is equal to
1. 6119
2. 7471
3. 4941
4. 6090
Choice A
6119
Correct: 17.86%
Incorrect: 10.91%
Unattempted: 71.23%
CAT 2021 Slot 1 - QA
How many three-digit numbers are greater than 100 and increase by 198 when the three digits are arranged in the reverse order?
70
Correct: 12.29%
Incorrect: 38.8%
Unattempted: 48.91%
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?
Correct Answer: 6
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
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
Correct Answer: 2704
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?
Correct Answer: 315
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?
Correct Answer: 21
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
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
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
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]
7
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
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
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]
40
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]
3
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
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]
3

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.

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
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
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
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
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.
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
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
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.

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
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
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)
IPMAT 2020 Sample Paper - IPM Rohtak Quants
18800 / 470 / 20
1. 1
2. 2
3. 3
4. 4
Choice B
2
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
IPMAT 2019 Question Paper - IPM Indore Quants
Assume that all positive integers are written down consecutively from left to right as in 1234567891011...... The 6389^th digit in this sequence is
4
IPMAT 2019 Question Paper - IPM Indore Quants
The number of pairs of integers whose sums are equal to their products is
2
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