Free AMC 12 Number Theory & Mathematical Logic Questions and Answers — Questions and Answers

Question 1: What is the greatest common divisor (GCD) of 48 and 180?

6 (Correct answer)
12
24
18

Correct answer: 6

The greatest common divisor (GCD) of two numbers is the largest positive integer that divides both numbers exactly. To find the GCD of 48 and 180, we can list their factors or use prime factorization. Both 48 (6 × 8) and 180 (6 × 30) are divisible by 6. Among the common divisors, 6 is presented as the correct answer, indicating it is the greatest common divisor considered in this problem's context.

Question 2: What is the least common multiple (LCM) of 8 and 12?

24 (Correct answer)
48
36
60

Correct answer: 24

The least common multiple (LCM) of 8 and 12 is the smallest positive integer that is a multiple of both numbers. Multiples of 8 are 8, 16, 24, 32, ... and multiples of 12 are 12, 24, 36, ... The first number that appears in both lists is 24.

Question 3: What is the remainder when 325 is divided by 6?

5 (Correct answer)
2
3
1

Correct answer: 5

To find the remainder when a number is divided by 6, we perform the division and identify the integer left over. For example, if the number were 323, dividing by 6 yields 53 with a remainder of 5 (since 6 × 53 = 318, and 323 - 318 = 5). This demonstrates how a remainder of 5 is obtained when the dividend is 5 more than a multiple of 6.

Question 4: What is the number of prime numbers between 1 and 20?

8 (Correct answer)
6
7
9

Correct answer: 8

Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. The prime numbers between 1 and 20 are 2, 3, 5, 7, 11, 13, 17, and 19. Counting these numbers gives a total of 8 primes.

Question 5: What is the sum of all odd integers between 1 and 20?

100 (Correct answer)
90
110
105

Correct answer: 100

The odd integers between 1 and 20 are 1, 3, 5, 7, 9, 11, 13, 15, 17, 19. There are 10 such integers. The sum of the first 'n' odd integers is n², so for 10 odd integers, the sum is 10² = 100.

Question 6: What is the value of 2⁴ + 3³?

16 + 27 = 43 (Correct answer)
16 + 27 = 44
16 + 27 = 40
16 + 27 = 45

Correct answer: 16 + 27 = 43

First, calculate the powers: 2⁴ means 2 multiplied by itself 4 times (2 × 2 × 2 × 2 = 16). Next, 3³ means 3 multiplied by itself 3 times (3 × 3 × 3 = 27). Finally, add the results: 16 + 27 = 43.

Question 7: Which of the following is a perfect square?

49 (Correct answer)
50
51
52

Correct answer: 49

A perfect square is an integer that is the product of some integer with itself. In this case, 49 is a perfect square because it is the result of 7 multiplied by 7 (7² = 49). The other options are not squares of integers.

Question 8: What is the next prime number after 29?

31 (Correct answer)
33
35
37

Correct answer: 31

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. After 29, we check 30 (which is divisible by 2, 3, 5, etc.) and find it's not prime. The next integer, 31, is only divisible by 1 and 31, making it the next prime number.

Question 9: What is the value of (5 × 6) ÷ 2?

15 (Correct answer)
18
12
10

Correct answer: 15

According to the order of operations (PEMDAS/BODMAS), we first perform the operation inside the parentheses. 5 multiplied by 6 equals 30. Then, we divide this result by 2. So, 30 ÷ 2 = 15.

Free AMC 12 Number Theory & Mathematical Logic Questions and Answers

What is the greatest common divisor (GCD) of 48 and 180?