GCD & LCM — Practice Set

Bank code: NUM-GCL · Section: Number Properties · 30 questions — Easy 5 · Medium 8 · Hard 12 · Extreme 5

GCD takes the lowest powers of the shared primes and answers grouping questions; LCM takes the highest powers of every prime and answers coinciding questions; and for two numbers GCD x LCM = the product.

Every question below is live in the question bank under the ID shown — the sheet and the portal are the same questions. Attempt a level with the clock running, then check Part C.

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Part A — Questions

Quantitative Comparison — the four choices are always the same, so they are not reprinted: (A) Column A is greater · (B) Column B is greater · (C) The two quantities are equal · (D) The relationship cannot be determined from the information given

Level 1 · Easy — 5 questions · ~4 min

warm-up — these must be automatic

Q1 · NUM-GCL-001 · MCQ · 45s

What is the GCD of 12 and 18?

(A) 2
(B) 3
(C) 6
(D) 9
(E) 36

Q2 · NUM-GCL-011 · Numeric Entry · 50s

Bus A departs every 6 minutes and Bus B departs every 9 minutes. Both depart at 8:00 AM. How many minutes after 8:00 AM do they next depart at the same time? Enter your answer as a number.

Numeric entry — write the number.

Q3 · NUM-GCL-012 · MCQ · 55s

a and b are positive integers with LCM(a, b) = a. Which of the following must be true?

(A) a = b
(B) a > b
(C) b divides a
(D) a divides b
(E) GCD(a, b) = 1

Q4 · NUM-GCL-013 · Select all that apply · 55s

Which of the following are common divisors of 30 and 45? Select all that apply.

(A) 1
(B) 3
(C) 5
(D) 15
(E) 30

Q5 · NUM-GCL-014 · QC · 50s

p and q are distinct prime numbers.

Column A: LCM(p, q) Column B: p x q

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Level 2 · Medium — 8 questions · ~10 min

two or three steps, one planted trap each

Q6 · NUM-GCL-028 · MCQ · 80s

a = 2^5 x 3^2 x 7 and b = 2^3 x 3^4 x 5. What is GCD(a, b)?

(A) 2^3 x 3^2
(B) 2^5 x 3^4
(C) 2^5 x 3^4 x 5 x 7
(D) 2^3 x 3^2 x 5 x 7
(E) 2 x 3

Q7 · NUM-GCL-024 · MCQ · 75s

A rectangular floor measuring 84 feet by 126 feet is to be covered exactly by identical square tiles, with no tile cut and no gaps left. What is the largest possible side length, in feet, of each tile?

(A) 6
(B) 14
(C) 21
(D) 42
(E) 84

Q8 · NUM-GCL-026 · MCQ · 85s

The positive integers a and b satisfy GCD(a, b) = 4, and neither a nor b exceeds 20. What is the greatest possible value of LCM(a, b)?

(A) 20
(B) 60
(C) 80
(D) 100
(E) 320

Q9 · NUM-GCL-019 · QC · 65s

n is a positive integer.

Column A: GCD(n, n + 1) Column B: 1

Q10 · NUM-GCL-035 · QC · 80s

m and n are positive integers with m > n > 1.

Column A: GCD(m, n) Column B: GCD(m - n, n)

Q11 · NUM-GCL-021 · Numeric Entry · 70s

What is the LCM of 12, 15 and 20? Enter your answer as a number.

Numeric entry — write the number.

Q12 · NUM-GCL-029 · Numeric Entry · 80s

Three ropes of lengths 24 m, 36 m and 48 m are to be cut into equal pieces of the greatest possible length, with no rope left over. How many pieces are produced in total? Enter your answer as a number.

Numeric entry — write the number.

Q13 · NUM-GCL-033 · Select all that apply · 85s

a and b are positive integers. Which of the following statements are true for EVERY such a and b? Select all that apply.

(A) LCM(a, b) >= a
(B) LCM(a, b) <= a x b
(C) LCM(a, b) = a x b
(D) LCM(a, b) is a multiple of GCD(a, b)
(E) GCD(a, b) < LCM(a, b)

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Level 3 · Hard — 12 questions · ~20 min

where 162+ is won or lost

Q14 · NUM-GCL-030 · MCQ · 100s

a and b are positive integers and GCD(a, b) = d. Which of the following must be true?

(A) d^2 divides a + b
(B) a/d and b/d are both prime
(C) a x b = d x LCM(a, b)
(D) GCD(a/d, b/d) = d
(E) d divides a - b only when a > b

Q15 · NUM-GCL-039 · MCQ · 100s

a, b and c are positive integers with GCD(a, b) = 12 and GCD(b, c) = 15. What is the largest possible value of GCD(a, b, c)?

(A) 1
(B) 3
(C) 4
(D) 6
(E) 60

Q16 · NUM-GCL-042 · MCQ · 100s

The LCM of two positive integers is 5 times their GCD, and the two integers sum to 36. What are the two integers?

(A) 6 and 30
(B) 12 and 24
(C) 15 and 21
(D) 18 and 18
(E) 4 and 32

Q17 · NUM-GCL-051 · MCQ · 100s

A store restocks shelves every 12 days, makes deliveries every 8 days and takes inventory every 15 days. All three happen today, which is day 0. On how many of the next 1000 days - that is, days 1 through 1000 - do all three happen on the same day?

(A) 4
(B) 8
(C) 9
(D) 16
(E) 41

Q18 · NUM-GCL-061 · MCQ · 105s

What is the greatest integer that divides 400, 435 and 541, leaving remainders 9, 10 and 14 respectively?

(A) 5
(B) 17
(C) 23
(D) 31
(E) 85

Q19 · NUM-GCL-062 · MCQ · 105s

p and q are distinct prime numbers, each greater than 3. Let a = 12 p^3 q^2 and b = 18 p^2 q^4. What is LCM(a, b) / GCD(a, b)?

(A) 6 p q^2
(B) 3 p q^2
(C) 36 p q^2
(D) 6 p^5 q^6
(E) 216 p^5 q^6

Q20 · NUM-GCL-049 · QC · 105s

p is a prime number greater than 3.

Column A: GCD(p^2 - 1, 24) Column B: 24

Q21 · NUM-GCL-053 · QC · 110s

a = 2^3 x 3^2 x 5, b = 2^2 x 3 x 5^2 x 7 and c = 2 x 3^3 x 5.

Column A: GCD(a, b, c) x LCM(a, b, c) Column B: a x b x c

Q22 · NUM-GCL-040 · Numeric Entry · 100s

For how many positive integers L less than 300 do there exist positive integers a and b with GCD(a, b) = 12 and LCM(a, b) = L? Enter your answer as a number.

Numeric entry — write the number.

Q23 · NUM-GCL-052 · Numeric Entry · 110s

How many positive integers n with n <= 500 satisfy GCD(n, 500) = 20? Enter your answer as a number.

Numeric entry — write the number.

Q24 · NUM-GCL-043 · Select all that apply · 100s

n is a positive integer and g = GCD(n, 12). Which of the following must be a multiple of g? Select all that apply.

(A) n + 12
(B) n - 6, where n > 6
(C) 12n
(D) n^2
(E) n + 18

Q25 · NUM-GCL-050 · Select all that apply · 95s

a and b are positive integers. Which of the following expressions equal LCM(a, b) for every such a and b? Select all that apply.

(A) (a x b) / GCD(a, b)
(B) a x (b / GCD(a, b))
(C) (a / GCD(a, b)) x b
(D) (a x b) - GCD(a, b)
(E) GCD(a^2, b^2) / GCD(a, b)

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Level 4 · Extreme — 5 questions · ~10 min

165+ — expect to need the insight, not the grind

Q26 · NUM-GCL-057 · MCQ · 120s

a and b are positive integers with a x b = 1440. What is the greatest possible value of GCD(a, b)?

(A) 12
(B) 24
(C) 36
(D) 48
(E) 120

Q27 · NUM-GCL-055 · Numeric Entry · 115s

For a positive integer n, let f(n) = LCM(1, 2, 3, ..., n). What is the value of f(10) / f(9)? Enter your answer as a number.

Numeric entry — write the number.

Q28 · NUM-GCL-058 · Numeric Entry · 125s

How many ordered pairs of positive integers (a, b) satisfy GCD(a, b) = 6 and LCM(a, b) = 1260? Enter your answer as a number.

Numeric entry — write the number.

Q29 · NUM-GCL-056 · QC · 120s

a, b and c are positive integers with GCD(a, b) = GCD(b, c) = GCD(a, c) = 1.

Column A: LCM(a, b, c) Column B: a x b x c

Q30 · NUM-GCL-060 · Select all that apply · 115s

For each positive integer n that divides 100, let S(n) = GCD(n, 100) + LCM(n, 100). Which of the following values of S(n) are achievable? Select all that apply.

(A) 101
(B) 108
(C) 125
(D) 180
(E) 200

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