Let n be a natural number greater than or equal to 3 such that n-3 is a multiple of 4. Show that n squared 6n 5 is a multiple of 16.

Answer 1

We know that n - 3 is a multiple of 4. Let's express n - 3 as 4k, where k is a positive integer.

Adding 3 to both sides, we have n = 4k + 3.

Now, let's substitute this expression for n in the expression n² + 6n + 5:

n² + 6n + 5 = (4k + 3)² + 6(4k + 3) + 5.

Expanding and simplifying, we have:

n² + 6n + 5 = 16k² + 24k + 9 + 24k + 18 + 5
= 16k² + 48k + 32
= 16(k² + 3k + 2).

Since k² + 3k + 2 is an integer, let's call it m.

Therefore, n² + 6n + 5 is equal to 16m, showing that it is a multiple of 16.

Thus, we have shown that if n - 3 is a multiple of 4, then n² + 6n + 5 is a multiple of 16.