If 49^ n +16 n+P is divisible by 64 for all n N , then the least negative integral value of P is

Step-by-step solution

Let f(n) = 49^ n + 16n + P . We are given that f(n) is divisible by 64 for all n N . For n = 1 , f(1) = 49¹ + 16(1) + P = 49 + 16 + P = 65 + P . For f(1) to be divisible by 64, 65 + P = 64k for some integer k . For n = 2 , f(2) = 49² + 16(2) + P = 2401 + 32 + P = 2433 + P . For f(2) to be divisible by 64, 2433 + P = 64m for some integer m . Subtracting the two conditions: (2433 + P) - (65 + P) = 64m - 64k , which gives 2368 = 64(m - k) . Since 2368 / 64 = 37 , the condition is consistent for all n . From 65 + P = 6