Q: Prove if a|k, b|k, then lcm(a,b)|k.

(Hint: LCM of non-zero intergers a and b is the smallest postive integer m such that a|m, b|m)

Please help!

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- June 5th 2007, 10:14 AMtttcomraderLCM problem
Q: Prove if a|k, b|k, then lcm(a,b)|k.

(Hint: LCM of non-zero intergers a and b is the smallest postive integer m such that a|m, b|m)

Please help! - June 5th 2007, 10:19 AMThePerfectHacker
- June 5th 2007, 11:07 AMThePerfectHackerQuote:

Originally Posted by**Neils Henrik Abel**

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Let then that means:

Now, by the above statements,

**Definition:**Let . For we define so that .

**Theorem:**Let and . Then .

**Proof:**Again, just follow the definitions.

Substitute second equation into first,

Since we have,

- June 5th 2007, 12:13 PMtttcomrader
The k that you use in your proof with c = ak, is that the same k as the one given in the problem?

And how does the lcm = m fit in this? - June 5th 2007, 12:27 PMThePerfectHacker
- June 5th 2007, 01:01 PMtttcomrader
I understand this proof, but would you mind proving the theorem? I don't understand how a|bc and c|a would implies (a/c)|b.

thanks so much! - June 5th 2007, 01:40 PMThePerfectHacker
- June 5th 2007, 01:43 PMtttcomrader
Oh, I missed that last part, now I fully understand it, thanks!