Induction Proof Problem

Question

I'm having some problems with this one.

Let $p,q,k\in \mathbb{Z}$. If $p$ and $q$ are divisible by $k$, then $p+q$ is divisible by $k$.

Answer 1

You don't need induction for this.

If p and q are divisible by k, then p = kx and q = ky for some integers x and y. Thus, p + q = kx + ky = k(x+y), which shows p + q is divisible by k.

Answer 2

You asked for induction. This is induction.

Base case $p = q = 0$

Then $p + q = 0$ and $k|0$.

Induction step: $p$ and $q$ are multiples of $k$ and $k|p+q$

Then $p + q = mk$ for some $m$.

So $p + q \pm k= (m\pm 1)k$ so $k|p + q \pm k$.

So $p + q \pm k = p + (q \pm k) = (p \pm k) + q$

So $k|p + (q\pm k)$ and $k|(p \pm k) + q$.

So $k|P + Q$ for $P$ and $Q$ being any multiple of $k$.

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but seriously, induction is probably not the way to go.