Which positive integers less than 12 are relatively prime to 12?

To determine which positive integers less than 12 are relatively prime to 12, we need to identify numbers that share no common factors with 12 other than 1.

The prime factorization of 12 is 2² × 3. This means any number that has 2 or 3 as a factor cannot be relatively prime to 12.

Now, let’s examine the positive integers less than 12: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11.

• 1: GCD(1, 12) = 1 (relatively prime)
• 2: GCD(2, 12) = 2 (not relatively prime)
• 3: GCD(3, 12) = 3 (not relatively prime)
• 4: GCD(4, 12) = 4 (not relatively prime)
• 5: GCD(5, 12) = 1 (relatively prime)
• 6: GCD(6, 12) = 6 (not relatively prime)
• 7: GCD(7, 12) = 1 (relatively prime)
• 8: GCD(8, 12) = 4 (not relatively prime)
• 9: GCD(9, 12) = 3 (not relatively prime)
• 10: GCD(10, 12) = 2 (not relatively prime)
• 11: GCD(11, 12) = 1 (relatively prime)

From this evaluation, the positive integers less than 12 that are relatively prime to 12 are: 1, 5, 7, and 11.