AMC 8 Number Theory & Divisibility 2 — Questions and Answers
Question 1: What is the value of 2¹⁰?
- 512
- 1000
- 1024 (Correct answer)
- 2048
Correct answer: 1024
2¹⁰ = 1024, which can be computed by doubling from 1 ten times.
Question 2: Which of the following numbers is divisible by both 4 and 6?
- 14
- 18
- 24 (Correct answer)
- 28
Correct answer: 24
24 is divisible by both 4 (24÷4 = 6) and 6 (24÷6 = 4); the others fail at least one test.
Question 3: What is the remainder when 100 is divided by 7?
- 1
- 2 (Correct answer)
- 3
- 6
Correct answer: 2
100 = 7 × 14 + 2, so the remainder is 2.
Question 4: How many two-digit positive integers are divisible by 9?
- 9
- 10 (Correct answer)
- 11
- 12
Correct answer: 10
The two-digit multiples of 9 are 18, 27, 36, 45, 54, 63, 72, 81, 90, 99 — exactly 10 numbers.
Question 5: What digit d makes the three-digit number 4d2 divisible by 3?
- 0 (Correct answer)
- 1
- 5
- 7
Correct answer: 0
For divisibility by 3, the digit sum 4+0+2 = 6 is divisible by 3, so d = 0 works.
Question 6: What is the units digit of 7²⁰?
- 1 (Correct answer)
- 3
- 7
- 9
Correct answer: 1
The units digits of powers of 7 cycle as 7, 9, 3, 1 (period 4); since 20 is divisible by 4, 7²⁰ ends in 1.
AMC 8 Number Theory & Divisibility 2
What is the value of 2¹⁰?