# Solving two equations in a creative way

Given here are two equations.

Assuming "a" and "b" are not zero or infinity and using some creative thinking,

prove that "b" is two times "a".

7a x 3a x 5a = 10a

13b x 5b = 15b

Suppose we treat "a" and "b" as

postfix operators.

In math, a common one is

! for factorial.

If we set this as "a", then "b" is two times "a", i.e.,

!! for double factorial. For any natural number "n", n!! is defined recursively as follows:
0!! = 1!! = 1,
n!! = n x (n-2)!! for n>1.

Indeed,

7! x 3! x 5! = 7! x 6 x 120 = 7! x 8 x 9 x 10 = 10!

and

13!! x 5!! = 13!! x 15 = 15!!

• @DrD That answer doesn't look like a proof that b is two times a. It looks like "I assumed that b was two times a, and I have not encountered a contradiction so far". Perhaps the question could be rephrased "Fill in values for a and b such that b is two times a" rather than "Prove that b is two times a".
– Stef
Apr 5, 2022 at 10:08
• @Stef I was chosing between either "show" or "prove". As you can see from the answer noedene showed b is indeed two times a. Appreciate your comment.
– DrD
Apr 5, 2022 at 12:03
• @Stef Agreed, there is no "proof" possible without defining precisely what is allowed. Replacing "prove" by the synonym "show" doesn't change this either. It is more like, "Find "a" and "b" that work. Hint: "b" is two times "a"." Apr 5, 2022 at 15:56