(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that for all integers x >= 8, x can be written in the form 3m + 5n, where m and n are non-negative integers.

2. Relevant equations

3. The attempt at a solution

Proof by induction on n that every integer n >= 8 can be expressed as n = 5x + 3y, with some integers x and y.

Let n = 8. Then n = 8 = 5(1) + 3(1), so the proposition is true for the base case.

Suppose the proposition is true for some number integer n = k > 8, i.e. k = 5x + 3y, for integers x and y. Consider the case when n = k + 1.

Then we have

k + 1 = 5x + 3y + 1

= 5x + 3y + 1 + 5 - 5

= 5x - 5 + 3y + 6

= 5(x - 1) + 3(y + 2).

Since the proposition is true for the base case and it being true for n = k implies

that it is true for n = k + 1, then n = 5x + 3y for some integers x and y.

I think that's almost it, but what about showing that m and n will never have to be negative?

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# Homework Help: K = 5m + 3n

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