GRE Arithmetic and Number Properties 2 — Questions and Answers
Question 1: What is the units digit of 7^23?
- 1
- 3 (Correct answer)
- 7
- 9
Correct answer: 3
Powers of 7 cycle 7,9,3,1 with period 4; 23 mod 4 = 3, giving units digit 3.
Question 2: If n is an integer, which expression is always even?
- n^2 + n (Correct answer)
- n^2 + 1
- 2n + 1
- n^2 - n + 1
Correct answer: n^2 + n
n^2 + n = n(n+1) is a product of two consecutive integers, so it is always even.
Question 3: What is the greatest common divisor (GCD) of 84 and 120?
- 6
- 12 (Correct answer)
- 24
- 42
Correct answer: 12
84 = 2^2·3·7 and 120 = 2^3·3·5, so the GCD is 2^2·3 = 12.
Question 4: How many distinct prime factors does 360 have?
- 2
- 3 (Correct answer)
- 4
- 5
Correct answer: 3
360 = 2^3·3^2·5, which has the three distinct primes 2, 3, and 5.
Question 5: If the remainder when x is divided by 5 is 3, what is the remainder when 2x is divided by 5?
- 1 (Correct answer)
- 2
- 3
- 4
Correct answer: 1
2x ≡ 2·3 = 6 ≡ 1 (mod 5), so the remainder is 1.
Question 6: Which of the following is the least common multiple (LCM) of 6, 8, and 9?
- 36
- 48
- 72 (Correct answer)
- 144
Correct answer: 72
Taking highest prime powers 2^3·3^2 = 72.
Question 7: If a positive integer has exactly 3 divisors, it must be which of the following?
- A prime number
- The square of a prime (Correct answer)
- A perfect cube
- An even number
Correct answer: The square of a prime
A number with exactly 3 divisors is p^2 for a prime p, giving divisors 1, p, and p^2.
GRE Arithmetic and Number Properties 2
What is the units digit of 7^23?