GMAT Hard Math Questions | Arithmetic

Must solve GMAT practice questions in arithmetic. These GMAT hard math questions include both problem solving and data sufficiency questions. The following topics are covered in the GMAT quant section from arithmetic:

1. Number Properties & Theory
2. Ratio and Proportions
3. Percents
4. Powers and Roots of Numbers
5. Descriptive Statistics
6. Sets
7. Counting Methods: Permutation & Combination

1. 700 800 Level GMAT Sample Question | Probability

   If two distinct integers a and b are picked from {1, 2, 3, 4, .... 100} and multiplied, what is the probability that the resulting number has EXACTLY 3 factors?

   1. \\frac{4}{25 \times 99})
   2. \\frac{2}{25 \times 99})
   3. \\frac{8}{25 \times 99})
   4. \\frac{16}{25 \times 99})
   5. \\frac{32}{25 \times 99})

   Correct AnswerChoice B
   2/(25 × 99)

   Solution Video Solution

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2. GMAT Hard Math Question | Number Properties DS Practice

   What is the remainder when the positive integer x is divided by 6?

   Statement 1: When x is divided by 7, the remainder is 5.
   Statement 2: When x is divided by 9, the remainder is 3.

   Correct AnswerChoice E
   Statements Together NOT Sufficient

   Solution Video Solution

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3. GMAT 700 Level Sample Question in Statistics

   Consider a set S = {2, 4, 6, 8, x, y} with distinct elements. If x and y are both prime numbers and 0 < x < 40 and 0 < y < 40, which of the following MUST be true?

   I. The maximum possible range of the set is greater than 33.
   II. The median can never be an even number.
   III. If y = 37, the average of the set will be greater than the median.

   1. I only
   2. I and II only
   3. I and III only
   4. III only
   5. I, II, and III

   Correct AnswerChoice E
   All 3 statements are correct.

   Solution Video Solution

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4. GMAT Challenging Question | Statistics Problem Solving

   Three positive integers a, b, and c are such that their average is 20 and a ≤ b ≤ c. If the median is (a + 11), what is the least possible value of c?

   1. 23
   2. 21
   3. 25
   4. 26
   5. 24

   Correct AnswerChoice C
   Least possible value for c is 25.

   Solution Video Solution

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5. GMAT Hard Math | Arithmetic | Permutation Combination Sample Question

   How many four-digit positive integers exist that contain the block 25 and are divisible by 75. (2250 and 2025 are two such numbers)?

   1. 90
   2. 63
   3. 34
   4. 87
   5. 62

   Correct AnswerChoice C
   34 such four-digit numbers exist.

   Solution Video Solution

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6. GMAT Challenging Math Question | Arithmetic | Number Properties DS Practice

   Is |x| > x?

   Statement 1: x² + y² = 4
   Statement 2: x³ + y² = 0

   Correct AnswerChoice C
   Statements Together are Sufficient.

   Solution Video Solution

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7. GMAT 700 800 Level Question | Coordinate Geometry & Permutation Combination

   Rectangle ABCD is constructed in the xy-plane so that sides AB and CD are parallel to the x-axis. Both the x and y coordinates of all four vertices of the rectangle are integers. How many rectangles can be constructed if x and y coordinates satisfy the inequality 11 < x < 29 and 5 ≤ y ≤ 13?

   1. 153
   2. 153C₄
   3. 4896
   4. 2448
   5. 5508

   Correct AnswerChoice C
   4896 rectangles can be constructed.

   Solution Video Solution

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8. GMAT Hard Math Questions | Arithmetic | Number Properties DS

   x is a two-digit positive integer. y is obtained by multiplying the tens place of x by 2. Is y > \\frac{x}{6})?

   Statement 1: 20 < x < 30
   Statement 2: y = 10

   Correct AnswerChoice B
   Statement 2 ALONE is sufficient.

   Solution Video Solution

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9. GMAT 700 Level Sample Question in Counting Methods

   149 is a 3-digit positive integer, product of whose digits is 1 × 4 × 9 = 36. How many 3-digit positive integers exist, product of whose digits is 36?

   1. 21
   2. 27
   3. 9
   4. 45
   5. 12

   Correct AnswerChoice A
   21 numbers exist.

   Solution Video Solution

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10. GMAT Hard Math Arithmetic | Permutation Combination - Selections | GMAT Problem Solving

    A student is required to solve 6 out of the 10 questions in a test. The questions are divided into two sections of 5 questions each. In how many ways can the student select the questions to solve if not more than 4 questions can be chosen from either section?

    1. 100
    2. 210
    3. 150
    4. 200
    5. 105

    Correct AnswerChoice D
    In 200 ways.

    Solution Video Solution

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11. GMAT 700 800 Level Question | Arithmetic | Counting Methods Problem Solving

    How many 6-digit numbers can be formed using the digits {1, 2, 3, ... 9} such that any digit that appears in such a number appears at least twice?

    1. 1809
    2. 7569
    3. 9369
    4. 8649
    5. 8289

    Correct AnswerChoice C
    9369 such numbers.

    Solution Video Solution

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12. GMAT Hard Math | Arithmetic | Number Properties Sample Question

    If y is the highest power of a number 'x' that can divide 101! without leaving a remainder, then for which among the following values of x will y be the highest?

    1. 111
    2. 462
    3. 74
    4. 33
    5. 210

    Correct AnswerChoice E
    When x = 210.

    Solution Video Solution

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13. GMAT Data Sufficiency | Challenging Number Properties & Inequalities Question

    Is x |x| = x²?

    Statement 1: x(1 – x²) < 0
    Statement 2: x(1 – x) < 0

    Correct AnswerChoice E
    Statements NOT sufficient.

    Solution Video Solution

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14. GMAT DS Sample Question | Statistics | Mean Median Range

    What is the range of 5 distinct single digit positive integers if their average is 5?

    Statement 1: Their median is 6.
    Statement 2: The average of the 3 largest among the 5 numbers is 7.

    Correct AnswerChoice C
    Statements Together Sufficient.

    Solution Video Solution

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15. GMAT Challenging Question | Arithmetic | Numbers & Progressions DS

    What is the range of 3 positive integers a, b, and c?

    Statement 1: 21a = 9b = 7c
    Statement 2: a + 8, b, and c, in that order are in AP.

    Correct AnswerChoice C
    Statements TOGETHER sufficient.

    Solution Video Solution

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16. GMAT Hard Math Questions | Arithmetic | Number Properties DS

    Is the twelve-digit positive integer a perfect square?

    Statement 1: The number comprises only the digits 0, 1, and 2, each written four times.
    Statement 2: The sum of the digits of the twelve-digit number is 12.

    Correct AnswerChoice D
    Statements INDEPENDENTLY sufficient.

    Solution Video Solution

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17. GMAT 650 Level Sample Question | Arithmetic DS | Number Properties

    If a and b are positive integers, is (a + b) prime?

    Statement 1: 13a = 43b
    Statement 2: 8a = 15b

    Correct AnswerChoice A
    Statement 1 ALONE sufficient.

    Solution Video Solution

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18. GMAT Hard Math Arithmetic - Counting Methods | GMAT Problem Solving

    What is the sum of all 3-digit positive integers such that all the digits of each of the number is even?

    1. 55,500
    2. 1300
    3. 44,400
    4. 247,275
    5. 54,400

    Correct AnswerChoice E
    54,400

    Solution Video Solution

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19. GMAT Challenging Math Question | Arithmetic | Number Properties - Remainders

    What is the least number that when divided by 44 leaves a remainder 31, when divided by 56 leaves a remainder 43, and when divided by 32 leaves a remainder 19?

    1. 2464
    2. 2477
    3. 2451
    4. 616
    5. 603

    Correct AnswerChoice C
    2451

    Solution Video Solution

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20. GMAT Data Sufficiency | Number Properties

    What is the value of (m + n) if m and n are positive integers?

    Statement 1: m² – n² = 105
    Statement 2: Neither m nor n is divisible by 8.

    Correct AnswerChoice E
    Data Insufficient.

    Solution Video Solution

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21. GMAT DS Sample Question | Arithmetic - Statistics | Mean, Median, and Range

    If sets A and B have n elements each, are the ranges of the two sets equal?

    Statement 1: Both sets are symmetric about their respective means.
    Statement 2: The median of both the sets is 50.

    Correct AnswerChoice E
    Data Insufficient

    Solution Video Solution

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22. GMAT Number Properties Practice | 650 Level Question | Factors Problem Solving

    What is the product of all the factors of the cube of a positive integer 'n' if the product of all the factors of square of n is n³?

    1. n⁴
    2. n⁶
    3. n⁹
    4. n⁵
    5. Cannot be determined

    Correct AnswerChoice B
    n⁶

    Solution Video Solution

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23. GMAT Arithmetic Question | Numbers Counting Methods Problem Solving

    How many even 3-digit positive integers with distinct digits are there?

    1. 400
    2. 320
    3. 328
    4. 356
    5. 408

    Correct AnswerChoice C
    328 integers.

    Solution Video Solution

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