GRE Number Theory and Integer Properties 3 — Questions and Answers
Question 1: What is the units digit of 3^47?
- 1
- 3
- 7 (Correct answer)
- 9
Correct answer: 7
Units digits of powers of 3 cycle 3,9,7,1 with period 4, and 47 ≡ 3 (mod 4) gives 7.
Question 2: How many integers between 1 and 100 inclusive are divisible by both 4 and 6?
- 6
- 8 (Correct answer)
- 12
- 16
Correct answer: 8
Divisibility by both means divisibility by LCM(4,6)=12, and there are 8 multiples of 12 up to 100.
Question 3: If a and b are integers with a·b = 0, which statement must be true?
- Both a and b are 0
- At least one of a or b is 0 (Correct answer)
- a = b
- Neither is 0
Correct answer: At least one of a or b is 0
A product of integers is zero only when at least one factor is zero.
Question 4: What is the sum of the first 10 positive even integers?
- 90
- 100
- 110 (Correct answer)
- 120
Correct answer: 110
The sum 2+4+...+20 equals 2(1+2+...+10) = 2·55 = 110.
Question 5: Which of the following numbers is divisible by 9?
- 4,572 (Correct answer)
- 3,461
- 5,283
- 6,140
Correct answer: 4,572
The digits of 4,572 sum to 18, which is divisible by 9.
Question 6: If p is a prime greater than 3, what is the remainder when p² is divided by 12?
- 1 (Correct answer)
- 3
- 5
- 7
Correct answer: 1
Any prime greater than 3 satisfies p² ≡ 1 (mod 12).
Question 7: How many trailing zeros does the number 25 × 16 × 5 contain?
- 1
- 2
- 3 (Correct answer)
- 4
Correct answer: 3
25×16×5 = 2000, which has 3 trailing zeros from three factors of 10.
GRE Number Theory and Integer Properties 3
What is the units digit of 3^47?