We “know” that when we multiply two numbers with two terms

a|b * a|c

where

b + c =10, and a is a single-digit number.

that

10a*10(a+1) + bc = (10a+b) (10a+c)

for example,

So what is going on here?

Obviously, by expansion:

and my conjecture is

Where b+c = 10

This is far closer, still short an a to divide on the right side but I suspect I will find it in the morning.

Thanks Simon… I am happier to accept that my algebra is weak than the other likely alternative.

Second try…

( I suspect there might be a proof down this path but I need to learn so more general facts about squares, sequences, and other bits and pieces )

I know that (a+b)^2 = a^2 +2ab+b^2 and for the case b=5 I can say

(a+b)^2 = a * (a+1) + b^2

but it is not obvious that I can say the same for a|b * a|c where b+c = 10.

If I assume that I can ( which is silly )

if I set these as equivalent

My rule was b + c = 10

So that lets me say a probably untrue and useless thing

@kinma Any tips? I assume my algebra has gone off the rails here but I’m not seeing it. Having another coffee .