To determine which of the given numbers has no prime factors in common with 60, we should first find the prime factors of 60 and then compare them with the prime factors of each option.
### Step-by-Step Solution:
1. Calculate the prime factors of 60:
- 60 can be factored as [tex]\(60 = 2^2 \times 3 \times 5\)[/tex].
- Therefore, the prime factors of 60 are [tex]\(\{2, 3, 5\}\)[/tex].
2. Determine the prime factors of each option:
- Option A: 9
- 9 can be factored as [tex]\(9 = 3^2\)[/tex].
- The prime factors of 9 are [tex]\(\{3\}\)[/tex].
- Option B: 24
- 24 can be factored as [tex]\(24 = 2^3 \times 3\)[/tex].
- The prime factors of 24 are [tex]\(\{2, 3\}\)[/tex].
- Option C: 49
- 49 can be factored as [tex]\(49 = 7^2\)[/tex].
- The prime factors of 49 are [tex]\(\{7\}\)[/tex].
- Option D: 69
- 69 can be factored as [tex]\(69 = 3 \times 23\)[/tex].
- The prime factors of 69 are [tex]\(\{3, 23\}\)[/tex].
3. Compare the prime factors of each option with the prime factors of 60:
- Option A: 9
- Common prime factors with 60: [tex]\(\{3\}\)[/tex].
- Option B: 24
- Common prime factors with 60: [tex]\(\{2, 3\}\)[/tex].
- Option C: 49
- Common prime factors with 60: [tex]\(\{\}\)[/tex] (no common prime factors).
- Option D: 69
- Common prime factors with 60: [tex]\(\{3\}\)[/tex].
4. Identify the option that has no common prime factors with 60:
- From the comparisons, we see that Option C (49) has no prime factors in common with 60.
Therefore, the correct option is:
Option C: 49
