Flipping coefficients

Number Theory Level 3

If $x$ is an integer such that $3x+4$ is divisible by 7, is $4x+3$ also divisible by 7?

Yes, always No, never Yes, but not all the time

3 solutions

Skye Rzym
Apr 9, 2017

We can write $4x+3 = 7x+7-(3x+4)$

Because $7x+7$ and $3x+4$ are divisible by 7, we can conclude that $\boxed{4x+3 \text{ is divisible by } 7}$

Quite a clean solution! :)

toshali mohapatra - 4 years, 1 month ago

Anirudh Sreekumar
Apr 9, 2017

$\left((3x+4)+(4x+3)\right)\pmod {7}=(7x+7)\pmod {7}\equiv 0\\ \implies \text{if } 3x+4\pmod{7}\equiv 0,\\\text{then } 4x+3\pmod{7}\equiv 0$

Md Zuhair
Apr 9, 2017

Here we go,

$7|3x+4$

So \(7|3x+4-7|)\ also.

Now $7|3x-3$

Hence $7|x-1$ .

Now see , $7|4x-4$

We can write as, $7|4x-4+7$

So $7|4x+3$ for all the integers $x$ in which $7|3x+4$ . $\Bigg(\boxed{PROVED}\Bigg)$

By the way, In some previous year,this was to be proved in RMO

Thank you