Divisibility if a is odd.

Question

Suppose that $a |(4 b+5 c)$ and $a |(2 b+2 c) .$ Prove that if $a$ is odd, then $a | b$ and $a | c$

So since $a |(2 b+2 c)$ this imply $a |(4 b+4 c)$ so $a |(4 b+5 c)$ after subtraction of two dividend we directly get $a | c$ but I am unable to get why it should divide $b$ if $a$ is odd.

lab bhattacharjee · Accepted Answer · 2019-10-17T09:06:50.020

3

$a$ must divide $4b+5c-2(2b+2c)=c$

$a$ must divide $2b+2c-2(c)=2b$

Now if $a$ is odd $a\nmid2,a$ must divide the odd part of $b$

edited Oct 17 '19 at 09:06

answered Oct 17 '19 at 09:05

lab bhattacharjee

274,582

1

Do you mean "The odd part of $2b$"? – Arthur Oct 17 '19 at 09:06
@Arthur, Actually of $b$ or of $2b$ – lab bhattacharjee Oct 17 '19 at 09:07
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Well, it is the $2$ in $2b$ we want to get rid of, not the even part of $b$. So I would think that "the odd part of $2b$" would be the most suitable term. – Arthur Oct 17 '19 at 09:08

Divisibility if a is odd.

1 Answers1