prove or disprove .if n is a non negative integer, then 5 divides 2 ⋅ 4^n...

prove or disprove .if n is a non negative integer, then 5 divides 2 ⋅ 4^n + 3⋅9^n.

Expert Solution

When, n = 0, we have, 2•4ⁿ + 3•9ⁿ = 2 + 3 = 5, divisible by 5

We have, 4 - 1 (mod 5)

So, 4ⁿ (-1)ⁿ (mod 5) , for all positive integer n

So, 2•4ⁿ 2•(-1)ⁿ (mod 5)

&, 9 - 1 (mod 5)

So, 9ⁿ (-1)ⁿ (mod 5) , for all positive integer n

So, 3•9ⁿ 3•(-1)ⁿ (mod 5)

So, 2•4ⁿ + 3•9ⁿ 2•(-1)ⁿ + 3•(-1)ⁿ (mod 5)

If, n is even, then, (-1)ⁿ = 1

Then, 2•4ⁿ + 3•9ⁿ 2 + 3 5 0 (mod 5)

So, 5 divides (2•4ⁿ + 3•9ⁿ) when n is even

If, n is odd, then, (-1)ⁿ = -1

Then, 2•4ⁿ + 3•9ⁿ - 2 - 5 - 5 0 (mod 5)

So, 5 divides (2•4ⁿ + 3•9ⁿ) when n is odd

So, 5 divides (2•4ⁿ + 3•9ⁿ) for all non-negative integer n.

Colby Messinger answered 1 year ago

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