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

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To determine which positive integers less than 12 are relatively prime to 12, we first need to understand what ‘relatively prime’ means. Two numbers are considered relatively prime (or coprime) if their greatest common divisor (GCD) is 1. In other words, they do not share any prime factors.

The positive integers less than 12 are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11.

Next, we find the prime factorization of 12:

  • 12 = 22 x 3

Now we will analyze each integer less than 12 to see if it shares any prime factors with 12:

  • 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 analysis, the positive integers less than 12 that are relatively prime to 12 are:

  • 1
  • 5
  • 7
  • 11

In summary, the integers 1, 5, 7, and 11 do not share any prime factors with 12, making them relatively prime to it.

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