We will call a natural number $n \in \mathbb{N}$ 'cool' if two natural numbers $t,k \in \mathbb{N}$ exist such that $n = 4t + 7k$
For example: 8 and 26 are cool because:
$ 4 \cdot 2 + 7 \cdot 0 = 8$
and
$4 \cdot 3 + 7 \cdot 2 = 26$
Q: Prove there is a finite amount of Non-cool natural numbers.
I thought about finding a natural number $N$ so that for all $n \geq N$ is 'cool', and to prove that with complete(strong) induction, however I a don't know how to continue from here..
Thank you!
We show that every number from $35$ onwards is cool. We simply consider $n-4t$ for $1 \leqslant t \leqslant 7$. As $4t$ leaves distinct remainders for each $t$, $n-4t$ will also leave distinct remainders when divided by $7$. As there are only $7$ possible remainders, one of these is divisible by $7$. Then, $n-4t=7k \implies n = 4t+7k$. Furthermore, we are guaranteed that $k$ is also a positive integer as $n-4t \geqslant n-4(7) \geqslant 35-28 \geqslant 7$.