Explanation:

- Statement: The number \(97^{30} - 14^{30}\) is to be checked for divisibility.

- Option 1: 37 but not 83

- If 97 mod 37 = 23 and 14 mod 37 = 14, then \(23^{30} - 14^{30}\) mod 37 needs evaluation.

- Power residue cycles could make it divisible by 37.

- Option 2: 83 but not 37

- If 97 mod 83 = 14 and 14 mod 83 = 14, then \(14^{30} - 14^{30}\) confirms it is divisible by 83.

- Option 3: Both 37 and 83

- Correct Answer: Both criteria from Option 1 and 2 hold true together.

- Option 4: Neither 37 nor 83

- Incorrect, both divisors are applicable.