Problem involving replacing numbers by a product

Question

I am working on the following problem:

Suppose we have a natural number $c$ that can be expressed as the sum of two other natural numbers, $a$ and $b$: $c = a+b$. If this happens, we say that $c$ is replaceable and we replace it by $a\cdot b$. The question is: starting from $c = 22$, can we get to 2021 using a finite number of these steps?

I have tried to find an invariant while performing the replacement step, but I have not been able to succeed. Can anyone give some hint on how to solve it? Maybe it does not even require an invariant, but I am stuck.

Answer 1

First observe the factorization 2021 = 43.47. Now, write 22 = 1 + 21, replace 22 by 1.21 = 21. Then, 21 = 6 + 15, replace 21 by 6.15 = 90. Finally write 90 = 43 + 47 and replace 90 by 43.47 = 2021.

Answer 2

There's also the not-much-thought approach:

Observe that we can replace $c$ with $c-1$.

So, we just need to reach a large enough number, then subtract 1 many times.
EG Converting $c > 4 $ to $ 2(c-2)$ as needed.

Notes

• Show that for any starting value $ c > 4$, we can then reach any other natural number
• What if we started with $ c = 1, 2, 3, 4$?