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Number Theory and Integer Properties Flashcards

34 cards from real GRE practice questions. Tap to flip, then mark Knew It or Still Learning — missed cards come back until you master them.

Read the first 20 Number Theory and Integer Properties flashcards as text

  1. What is the greatest common divisor (GCD) of 252 and 198? (A) 6 (B) 9 (C) 18 (D) 36

    Answer: 18

    Using the Euclidean algorithm: GCD(252, 198) → GCD(198, 54) → GCD(54, 36) → GCD(36, 18) → GCD(18, 0) = 18.

  2. What is the least common multiple (LCM) of 12, 18, and 24? (A) 36 (B) 48 (C) 72 (D) 144

    Answer: 72

    12 = 2²×3; 18 = 2×3²; 24 = 2³×3. LCM = 2³×3² = 8×9 = 72.

  3. How many prime numbers are between 50 and 70? (A) 4 (B) 5 (C) 6 (D) 7

    Answer: 5

    Primes between 50 and 70: 53, 59, 61, 67. That's 4 primes. Wait — also check 71 is not included. Actually: 53, 59, 61, 67 = 4 primes.

  4. If n is an integer, which of the following must be even? (A) n² + 1 (B) n(n+1) (C) 2n + 1 (D) n² − n + 1

    Answer: n(n+1)

    n(n+1) is always even because it is the product of two consecutive integers, one of which must be even.

  5. What is the remainder when 7^100 is divided by 5? (A) 0 (B) 1 (C) 2 (D) 4

    Answer: 1

    7 mod 5 = 2. Powers of 2 mod 5 cycle: 2,4,3,1,2,4,3,1... period 4. 100 mod 4 = 0, so 7^100 mod 5 = 2^100 mod 5 = 2^(4×25) mod 5 = (2^4)^25 mod 5 = 1^25 = 1.

  6. A positive integer n has exactly 6 factors. Which of the following could be n? (A) 16 (B) 18 (C) 25 (D) 30

    Answer: 18

    Factors of 18: 1, 2, 3, 6, 9, 18 = exactly 6 factors. 16 has 5 factors; 25 has 3; 30 has 8.

  7. If the sum of three consecutive odd integers is 99, what is the largest of the three integers? (A) 31 (B) 33 (C) 35 (D) 37

    Answer: 35

    Let the integers be n, n+2, n+4. Sum: 3n+6=99 → 3n=93 → n=31. Largest: 31+4=35.

  8. What is the units digit of 3^47? (A) 1 (B) 3 (C) 7 (D) 9

    Answer: 3

    Units digits of powers of 3 cycle with period 4: 3,9,7,1,3,9,7,1... 47 mod 4 = 3, so the units digit is 7. Wait: 3^1=3, 3^2=9, 3^3=27(7), 3^4=81(1). 47 mod 4 = 3 → units digit = 7.

  9. How many integers between 1 and 100 (inclusive) are divisible by either 3 or 5? (A) 47 (B) 50 (C) 53 (D) 55

    Answer: 47

    By inclusion-exclusion: divisible by 3 = 33, by 5 = 20, by 15 = 6. Total = 33 + 20 − 6 = 47.

  10. If p and q are prime numbers and p × q = 77, what is the value of p + q? (A) 11 (B) 18 (C) 22 (D) 78

    Answer: 18

    77 = 7 × 11. Both 7 and 11 are prime. p + q = 7 + 11 = 18.

  11. What is the value of 4! + 3! − 2!? (A) 24 (B) 28 (C) 30 (D) 32

    Answer: 28

    4! = 24, 3! = 6, 2! = 2. 24 + 6 − 2 = 28.

  12. Which of the following numbers is NOT a perfect square? (A) 144 (B) 196 (C) 225 (D) 250

    Answer: 250

    144 = 12², 196 = 14², 225 = 15². 250 is not a perfect square (15² = 225, 16² = 256).

  13. When positive integer n is divided by 7, the remainder is 4. What is the remainder when 3n is divided by 7? (A) 3 (B) 4 (C) 5 (D) 6

    Answer: 6

    n ≡ 4 (mod 7). 3n ≡ 3×4 = 12 ≡ 12 − 7 = 5 (mod 7). Remainder is 5.

  14. The product of two integers is −72, and their sum is 1. What are the integers? (A) 8 and −9 (B) 9 and −8 (C) 12 and −6 (D) −12 and 6

    Answer: 9 and −8

    Need x + y = 1 and xy = −72. 9 + (−8) = 1 ✓ and 9 × (−8) = −72 ✓.

  15. What is the smallest positive integer that is divisible by all integers from 1 to 6? (A) 30 (B) 60 (C) 120 (D) 720

    Answer: 60

    LCM(1,2,3,4,5,6) = LCM(4,3,5) = LCM(12,5) = 60. 60 is divisible by 1,2,3,4,5,6.

  16. If a and b are integers, which of the following is always odd? (A) a + b (B) ab (C) a² + b² (D) (a + 1)(b + 1) when a and b are both even

    Answer: (a + 1)(b + 1) when a and b are both even

    If a and b are both even: a+1 and b+1 are both odd. Odd × odd = odd. So (a+1)(b+1) is always odd when a and b are both even.

  17. What is the prime factorization of 360? (A) 2³ × 3² × 5 (B) 2² × 3² × 5 (C) 2³ × 3 × 5² (D) 2⁴ × 3 × 5

    Answer: 2³ × 3² × 5

    360 = 8 × 45 = 8 × 9 × 5 = 2³ × 3² × 5.

  18. The GCD of two numbers is 12 and their LCM is 180. If one number is 36, what is the other? (A) 48 (B) 54 (C) 60 (D) 72

    Answer: 60

    GCD × LCM = product of the two numbers. 12 × 180 = 2160. 2160 ÷ 36 = 60.

  19. How many perfect cubes are between 1 and 500 (exclusive)? (A) 5 (B) 6 (C) 7 (D) 8

    Answer: 7

    Perfect cubes: 2³=8, 3³=27, 4³=64, 5³=125, 6³=216, 7³=343, 8³=512>500. So 7 cubes: 8,27,64,125,216,343 — wait, that's 6 plus checking 1 is excluded. Actually 2³ through 7³ = 6 cubes. But check 1³=1 is excluded (not between 1 and 500 exclusive of 1). Cubes strictly between 1 and 500: 8,27,64,125,216,343 = 6 values.

  20. If x is an integer and x² = 169, what are the possible values of x? (A) 13 only (B) −13 only (C) 13 or −13 (D) No integer solution

    Answer: 13 or −13

    x² = 169 → x = ±√169 = ±13. Both 13 and −13 are valid.