If product is multiple of a number what information can we draw?

Asked Dec 11 '20 at 17:36

Active Dec 11 '20 at 17:44

Viewed 32 times

If a product is a multiple of $4$, I conclude that one of the numbers is a multiple of $2$ the following way:
If $$ab \equiv0\pmod 4$$ then $$ab = 4X = 2(2X) = 2Y$$ (where $Y = 2X$)
This implies that either $a$ or $b$ is a multiple of $2$ so $a$ or $b$ is even.

My question is if this approach is correct or there is a more "formal" way to do it.
Additionally what other conclusions can we draw by knowing that $ab$ is a multiple of $4$?

edited Dec 11 '20 at 17:44

Aiden Chow

2,849
10
32

asked Dec 11 '20 at 17:36

Jim

1,589

1

$4|ab\implies 2|ab\implies 2|a$ or $2|b$ by Euclid's lemma – J. W. Tanner Dec 11 '20 at 18:03
It's correct / complete only if you justify the claim that $,2\mid ab\Rightarrow 2\mid a,$ or $,2\mid b\ \ $ – Bill Dubuque Dec 11 '20 at 19:26
@BillDubuque: does that need proof? I thought it made sense – Jim Dec 11 '20 at 20:19
@BillDubuque: So the proof you mention is exactly Euclid's lemma? – Jim Dec 11 '20 at 21:20
What about 4? Can we deduce anything about multiples of 4? Or only about 2? – Jim Dec 11 '20 at 21:23
Yes, it needs proof. Yes, the linked statement is one form of Eudlid's Lemma. Applied again we can deduce that either both $a$ and $b$ are even or one of them is divisible by $4.\ \ $ – Bill Dubuque Dec 12 '20 at 02:17

If product is multiple of a number what information can we draw?

0 Answers0