To determine which positive integers less than 30 are relatively prime to 30, we need two numbers to be relatively prime. This means their greatest common divisor (GCD) is equal to 1.
Here’s a step-by-step process to find those numbers:
1. List all positive integers less than 30:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29.
2. Determine which pairs of these numbers with 30 have a GCD equal to 1:
- Check if 1 is relatively prime to 30. The GCD of 1 and 30 is 1.
- Check if 2 is relatively prime to 30. The GCD of 2 and 30 is 2 (since 2 divides 30).
- Check if 3 is relatively prime to 30. The GCD of 3 and 30 is 3 (since 3 divides 30).
- Check if 4 is relatively prime to 30. The GCD of 4 and 30 is 2.
- Check if 5 is relatively prime to 30. The GCD of 5 and 30 is 5 (since 5 divides 30).
- Check if 6 is relatively prime to 30. The GCD of 6 and 30 is 6.
- Check if 7 is relatively prime to 30. The GCD of 7 and 30 is 1.
- Continue similar checks for each number until 29.
By following this process, we can establish that the following numbers are relatively prime to 30:
1, 7, 11, 13, 17, 19, 23, and 29.
Thus, the positive integers less than 30 that are relatively prime to 30 are:
[tex]\[
\boxed{1, 7, 11, 13, 17, 19, 23, 29}
\][/tex]
