>>12208128Idk if this is a correct type of approach, but what do you think of this observation?
Consider a particular sum of 3s and/or 5s.
If the sum has at least two 3s in it, then you can replace the two 3s with a 5 and effectively subtract 1.
If it has at least two 5s, you can replace the two 5s with three 3s and subtract 1.
After either operation, the result will still be a sum of 3s and 5s.
Any (integer) sum of 3s and 5s greater than 8 will also itself have at least two 3s or two 5s.
So you can get to any particular number >=8 by just picking some greater number that is a sum of 3s and 5s, and counting downward in this manner until you reach the desired number.
So for any integer of the form 5a+3b, you can prove by induction that all smaller integers >=8 must also be representable in that form.
You can also count upward if you prefer, maybe most simply by just adding one three and then decrementing twice.
