Prove by induction that every integer n, n>=0 can be represented by 5a+7b for a, b in Z

Question

I feel that I could solve this question normally, especially given the closely-related question on this site, but I'm having trouble approaching it by induction.

I assume that 5a + 7b = n is true, then I try to show that it holds true for n+1. I'm assuming that I can't use the same a and b for the second equality, so I wrote it as 5a₂ +7b₂ = n+1.

From here, I substituted 5a + 7b for n, simplified and got to 5(a₂ - a) + 7(b₂ - b) = 1. I'm not sure if that's right or where to go from here

Hint: Try to find a multiple of $5$ and a multiple of $7$ that differ by $1$. — N. F. Taussig, Feb 16 '23 at 14:11
Other people already answered, but I just want to point out that induction relies on proving the fact for the base case first, so you should start with proving that you can find $a$ and $b$ such that $5a+7b=0$! — Carlos Santi Toledo, Feb 16 '23 at 14:17
@DietrichBurde I'm not sure if your comment is serious or not, but, since we need to prove the equality for $n\geq 0$, we actually need to consider the case $5a + 7b =0$. The exclamation mark was just that, not a factorial :) — Carlos Santi Toledo, Feb 16 '23 at 14:28
Does this answer your question? Show that every integer can be written in the form $5a + 7b$ for $a,b \in \mathbb{Z}$. One answer uses induction, too. — Dietrich Burde, Feb 16 '23 at 14:28
@CarlosSantiToledo Yes, you are right. I was distracted by $0!$ — Dietrich Burde, Feb 16 '23 at 14:29
A better question would have been to show that every integer $n\ge 24$ can be represented as $n=5a+7b$ for some nonnegative integers $a$ and $b$. — Alexander Burstein, Feb 17 '23 at 08:31

score 4 · Accepted Answer · answered Feb 16 '23 at 14:12

Maybe like this:

$$3 \cdot 5 - 2 \cdot 7 = 1$$

Hence, for every $n$:

$$ (3n) \cdot 5 + (-2n) \cdot 7 = n$$

Or is this too simple? :-)

If it must be done, using induction, you can do this:

Start with:

$$5a + 7b = n$$ and $$3 \cdot 5 + (-2) \cdot 7 = 1$$

Hence:

$$ (a + 3) \cdot 5 + (b - 2) \cdot 7 = n + 1$$

Prove by induction that every integer n, n>=0 can be represented by 5a+7b for a, b in Z

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