K = 23x + 7x, where x is an odd natural number. Which of the following need…

Concept: For any odd natural number n, and any integers a and b, the sum an + bn is always exactly divisible by (a + b). This divisibility identity holds only when the exponent is odd — it fails when the exponent is even.

Application: Here a = 23, b = 7, and the exponent x is given to be odd.

1. Apply the concept: since x is odd, K = 23x + 7x is divisible by (23 + 7) = 30.

2. Every divisor of 30 must therefore divide K for every odd x. Among the options, 10, 15, and 3 are all divisors of 30 — so these three always divide K.

3. 8 does not divide 30, so the identity above gives no guarantee that 8 divides K.

Cross-check: Test the smallest odd exponent, x = 1. K = 231 + 71 = 30. Checking each option against 30: 30 ÷ 10 = 3, 30 ÷ 15 = 2, 30 ÷ 3 = 10 — all exact — but 30 ÷ 8 = 3.75, which is not an integer. This confirms 8 fails to divide K at x = 1.

Result: 8 is the value that need not be a factor of K; 10, 15, and 3 are always factors of K for every odd x.