Divisible by 7 proof

[lapis flame]

I am having troble finishing this proof. I think I have started correctly but don't know where to go from here.

[sharp stag]

1. Wait patiently for a helper to come along.
2. Once someone helps you, say thank you and close the thread with:

+close

3. Feel free to nominate the person for helper of the week in #helper-nominations
4. Do not ping the mods, unless someone is breaking the rules.
5. If you're happy with the help you got here, and the server overall, you can contribute financially as well:

[sick swan]

what

ive always used last digit times 2 and minus the original

oh its the same thing

uhh try like 100a+10b+c

[cobalt kestrel]

if 7 | a + 5b, then 7 | n(a + 5b) for any n, see if you can find something convenient

[rare oxide]

Are you familiar with modular arithmetic?

[lapis flame]

@rare oxide yep, I tried to approach it but couldn't figure how to start it

like I get the last digit is n (mod 10) but like im not sure on the next steps

[cobalt kestrel]

e

[lapis flame]

like expand (10a+b)(a+5b)?

i cant think of a next step

[cobalt kestrel]

if 7 | a + 5b, then 7 | 10a + 50b

you want 7 | 10a + b

[lapis flame]

I have taken a different route that I think makes sense.

[cobalt kestrel]

i think this is a little bit easier but yeah

[jagged hemlock]

50b = 7*7b+b so 7|a+5b => 7|10a+50b => 7|10a+b

[lapis flame]

+close