I came across this as I was doing work for one of my classes. We just use this property, proved presumably in number theory, which we didn't need to take. Could someone help me? Prove that if $a$, $b$ are positive integers, and $m=\operatorname{lcm}(a,b)$, and if $s$ is a multiple of both $a$ and $b$, $s$ is a multiple of $m$. I tried representing each as a multiple, decomposing them, but I don't think I am getting it. I think I am missing some property. Thanks, especially if you could explain this using basic methods!
If $s$ is a multiple of both $a$ and $b$, then $s$ is a multiple of $\operatorname{lcm}(a,b)$
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abstract-algebra
elementary-number-theory
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0The property you want to prove is the way LCM is normally *defined*. (This is evident from the name: *least common multiple*.) What is your definition of the LCM? – 2012-01-06
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0@Srivatsan I think you're mistaken. The lcm is simply the smallest one among all common multiples. Therefore all others are _bigger than_ it. But "bigger than" is not the same as "multiples of". – 2012-01-06
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0@Michael That is a valid point, thanks for pointing it out. But I stand by my comment; see [this question](http://math.stackexchange.com/q/85565/13425) and [my answer](http://math.stackexchange.com/a/85576/13425) for an explanation. I do agree that under the definition I am imagining, there is an additional task of proving that the LCM/GCD *exists*. // Of course, it is likely that the OP defines it the way you do, but I would like to point out another point of view nevertheless. – 2012-01-06
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0@Michael Hardy: The disadvantage of this definition, is that if you do the same for gcd, then $gcd(0,0)$ is undefined (zero is a multiple of any integer). However, it is better to define it as 0, integers with $gcd$ then form a monoid etc. – 2012-01-06
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0@MichaelHardy For Euclidean domains one can indeed define LCM as a common divisor of least Euclidean value (= least absolute value in $\mathbb Z$), and dually for GCD. But generally one has no such structure available so one has to use the universal definition employing extremality w.r.t. divisibility, i.e. the property whose proof is sought in the question. See my [post here](http://math.stackexchange.com/a/88411/242) for more. – 2012-01-06