GRE Number Theory and Integer Properties 3

Question 1

What is the units digit of 3^47?

Accepted Answer

7

Answer

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?

Accepted Answer

8

Answer

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?

Accepted Answer

At least one of a or b is 0

Answer

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?

Accepted Answer

110

Answer

The sum 2+4+...+20 equals 2(1+2+...+10) = 2·55 = 110.

Question 5

Which of the following numbers is divisible by 9?

Accepted Answer

4,572

Answer

The digits of 4,572 sum to 18, which is divisible by 9.

GRE Practice Test