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Proof that LCM(c,d) divides LCM(a,b), known GCD(a,b) divides GCD(c,d)

  • Thread starter Ceci020
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  • #1
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** Homework Statement
1/
set S = set of the multiples of any two natural numbers a, b
S = {n in N such that a|n and b|n}

2/
Denote min(S) = LCM(a,b) = least common multiple of a and b

From previous result, I already proved that :
If a divides c and if b divides d, then GCD(a,b) divides GCD(c,d)

Now the question is to prove: LCM(a,b) divides LCM(c,d)


** My thoughts:
By definition of set S, x = LCM(a,b) satisfies the fact that a|n and b|n for n in N
I think, y = LCM(c,d) satisfies set S when c|m and d|m for m in N

But after that, I get confused on what to do next.
I think of trying to prove: LCM(c,d) = k * LCM(a,b), but I'm not sure if this is the right direction, or I need to do something else.

Could someone please give me on hints on what to do? Thank you in advance.:smile:
 

Answers and Replies

  • #2
Office_Shredder
Staff Emeritus
Science Advisor
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Hint: If a|n and b|n, then LCM(a,b)|n
 

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