Primes & Prime Factorization — Practice Set

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

1 is not prime and 2 is the only even prime - prime factorise first, because divisibility, GCD, LCM and factor counts all fall straight out of the exponents.

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-PRF-011 · Select all that apply · 55s

Which of the following numbers are prime? Select all that apply.

(A) 1
(B) 2
(C) 17
(D) 51
(E) 97

Q2 · NUM-PRF-003 · MCQ · 45s

What is the prime factorization of 60?

(A) 2 x 3 x 5
(B) 2^2 x 3 x 5
(C) 2^2 x 15
(D) 4 x 15
(E) 2 x 30

Q3 · NUM-PRF-012 · Numeric Entry · 45s

What is the largest prime factor of 84? Enter your answer as a number.

Numeric entry — write the number.

Q4 · NUM-PRF-009 · QC · 50s

Column A: The number of distinct prime factors of 30 Column B: The number of distinct prime factors of 42

Q5 · NUM-PRF-014 · MCQ · 55s

Two positive integers are relatively prime (coprime) when their greatest common divisor is 1. Which of the following pairs is relatively prime?

(A) 12 and 16
(B) 15 and 25
(C) 14 and 21
(D) 8 and 35
(E) 6 and 18

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Level 2 · Medium — 8 questions · ~10 min

two or three steps, one planted trap each

Q6 · NUM-PRF-016 · MCQ · 80s

What is the smallest positive integer k such that 1500k is a perfect cube?

(A) 6
(B) 15
(C) 18
(D) 30
(E) 54

Q7 · NUM-PRF-018 · Numeric Entry · 70s

What is the least common multiple of 14, 21, and 35? Enter your answer as a number.

Numeric entry — write the number.

Q8 · NUM-PRF-022 · Select all that apply · 85s

Which of the following statements about prime numbers are true? Select all that apply.

(A) Every prime greater than 3 is of the form 6k - 1 or 6k + 1 for some positive integer k
(B) The sum of two distinct primes is always even
(C) There are infinitely many primes
(D) If p and q are distinct primes, then GCD(p, q) = 1
(E) The product of any two primes is always odd

Q9 · NUM-PRF-023 · QC · 80s

p is a prime number and p > 2.

Column A: The remainder when p^2 is divided by 4 Column B: 1

Q10 · NUM-PRF-027 · MCQ · 70s

For positive integers a and b, GCD(a, b) = 12 and LCM(a, b) = 180. What is a x b?

(A) 144
(B) 168
(C) 192
(D) 2160
(E) 3600

Q11 · NUM-PRF-029 · Numeric Entry · 75s

What is the sum of the distinct prime factors of 1260? Enter your answer as a number.

Numeric entry — write the number.

Q12 · NUM-PRF-031 · Select all that apply · 85s

The positive integer n has exactly 3 positive factors. Which of the following must be true? Select all that apply.

(A) n is a perfect square
(B) n is prime
(C) n is odd
(D) sqrt(n) is prime
(E) The only factor of n strictly between 1 and n is sqrt(n)

Q13 · NUM-PRF-038 · MCQ · 75s

How many prime numbers p satisfy p^2 < 200?

(A) 4
(B) 6
(C) 7
(D) 8
(E) 14

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Level 3 · Hard — 12 questions · ~20 min

where 162+ is won or lost

Q14 · NUM-PRF-039 · MCQ · 95s

To establish that 881 is prime, it is enough to check that 881 is not divisible by any prime up to and including a certain prime. What is that prime?

(A) 23
(B) 29
(C) 31
(D) 439
(E) 881

Q15 · NUM-PRF-040 · Numeric Entry · 105s

What is the smallest positive integer that is divisible by each of 2, 3, 5 and 7 and has exactly 32 positive factors? Enter your answer as a number.

Numeric entry — write the number.

Q16 · NUM-PRF-042 · MCQ · 90s

For how many integers n with 2 <= n <= 50 is the product n(n - 1) divisible by 7?

(A) 7
(B) 8
(C) 13
(D) 14
(E) 15

Q17 · NUM-PRF-043 · Numeric Entry · 95s

What is the exponent of 3 in the prime factorization of 30! ? Enter your answer as a number.

Numeric entry — write the number.

Q18 · NUM-PRF-044 · QC · 90s

x and y are distinct prime numbers.

Column A: The sum of all positive factors of xy Column B: (x + 1)(y + 1)

Q19 · NUM-PRF-045 · MCQ · 110s

n = 2^a x 3^b x 5^c, where a, b and c are positive integers. If n is a perfect cube and n has exactly 64 positive factors, what is a + b + c?

(A) 6
(B) 9
(C) 11
(D) 12
(E) 18

Q20 · NUM-PRF-046 · Select all that apply · 100s

n is a positive integer. Which of the following expressions always have an odd number of positive factors? Select all that apply.

(A) n^2
(B) 4n^2
(C) 2n
(D) n^4
(E) 2n^2

Q21 · NUM-PRF-047 · Numeric Entry · 100s

What is the smallest positive integer n such that n! has at least 10 trailing zeros? Enter your answer as a number.

Numeric entry — write the number.

Q22 · NUM-PRF-048 · QC · 105s

p and q are distinct prime numbers, both greater than 2.

Column A: GCD(p + q, p x q) Column B: 2

Q23 · NUM-PRF-049 · MCQ · 110s

How many integers n with 1 <= n <= 100 have exactly 3 positive factors or exactly 4 positive factors?

(A) 30
(B) 32
(C) 34
(D) 36
(E) 38

Q24 · NUM-PRF-051 · Numeric Entry · 100s

How many positive divisors of 10,000 are perfect squares but not perfect cubes? Enter your answer as a number.

Numeric entry — write the number.

Q25 · NUM-PRF-052 · MCQ · 95s

What is the greatest integer m such that 3^m is a factor of 7! + 8! + 9! ?

(A) 2
(B) 4
(C) 5
(D) 6
(E) 8

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Level 4 · Extreme — 5 questions · ~10 min

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

Q26 · NUM-PRF-054 · MCQ · 115s

For a positive integer n, let f(n) be the number of trailing zeros of n!. Which of the following is NOT a possible value of f(n) for any positive integer n?

(A) 4
(B) 5
(C) 6
(D) 9
(E) 12

Q27 · NUM-PRF-056 · QC · 115s

n is an integer greater than 1, and p is the smallest prime factor of n.

Column A: p Column B: sqrt(n)

Q28 · NUM-PRF-058 · MCQ · 110s

p is a prime number and a is a positive integer. If p^3 divides a^2, which of the following must be true?

I. p^2 divides a

II. p^4 divides a^2

III. p divides a

(A) I only
(B) III only
(C) I and III only
(D) II and III only
(E) I, II and III

Q29 · NUM-PRF-059 · Select all that apply · 110s

Which of the following integers have exactly 4 positive factors? Select all that apply.

(A) 187
(B) 221
(C) 231
(D) 289
(E) 703

Q30 · NUM-PRF-060 · QC · 125s

p, q and r are distinct prime numbers with p < q < r.

Column A: The number of positive factors of p^2 x q^2 x r^2 Column B: The number of positive factors of p^2 x q x r x (p + q + r)

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