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# A little help please... Watch

1. Okay i'm having a bit of trouble starting this problem..

a, b, c are integers 0, ±1, ±2.... with a|b (a divides b) and a|c. Show that for any m, n (also integers), a|mb + nc ?

I'm having trouble working out the angle of attack on this beast. Other than trial and error (which proves nothing really), i can't see a way to prove it. Possibly, trying to contradict it?

I know that from the properties of divisability if a|b and a|c, then b|c.

Any help is appreciated. Cheers.
2. (Original post by ForeverOptimistic)

I know that from the properties of divisibility if a|b and a|c, then b|c.
This is false.

Counterexample: 2|6 and 2|4 but 6 does not divide 4.
3. (Original post by Get me off the £\?%!^@ computer)
This is false.

Counterexample: 2|6 and 2|4 but 6 does not divide 4.
You're right, i was getting carried away with myself. Any help then? I'm completely at a loss.

Cheers.
4. (Original post by ForeverOptimistic)
You're right, i was getting carried away with myself. Any help then? I'm completely at a loss.

Cheers.
b=ka for some integer k.

similar for c.

then mb+nc = mka+.... = a(.......)
5. (Original post by Get me off the £\?%!^@ computer)
b=ka for some integer k.

similar for c.

then mb+nc = mka+.... = a(.......)

So, c = pa for some integer p.

then mb + nc = mka + npa = a (mk + np)

And, m, k, n and p are all integers. Two integers added together make another, and if (mk + np) is an integer and two interers multiplied make another, then a(mk + np) must also be an integer.

So we've proved mb + nc is an integer, right?

To prove 'a| mb + nc' , could we say a = b/k and then (b/k) | mk(b/k) + np(b/k) = b | mb + npb. Hmm..not sure where to go there. I'm getting taken in circles, not sure how to finish the proof.
6. If mb+nc =integer*a then by definition a divides mb+nc.
7. (Original post by Get me off the £\?%!^@ computer)
If mb+nc =integer*a then by definition a divides mb+nc.
Cheers dude, i really appreciate it!
8. (Original post by ForeverOptimistic)
Cheers dude, i really appreciate it!
No problem. It keeps me out of trouble.
9. b=ka c=la,so mb=mka,nc=cla,both of them can be divided by a,so mb+nc can be divided by a

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