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0|0|0|||||Properties of GCD and LCM%3A Lemma proof|John|1|00:08:21|09/25/2010|I%27m confused by the proof of the Lemma for the Chinese Remainder Theorem on the page titled %22Properties of GCD and LCM.%22  The Lemma is %28or the part that%27s proven%29 %0D%0A%22lcm%28gcd%28N1%2C M%29%2C gcd%28N2%2C M%29%2C ...%2C gcd%28Nk%2C M%29%29 %3D gcd%28lcm%28N1%2C ...%2C Nk%29%2C M%29.%22  %0D%0A%0D%0AI know I%27m missing something%2C but I don%27t see how it was actaully proven.  The proof given forces me to conclude only this%3A that if a prime p dvides the left hand side%2C then it divides the right hand side--and vice versa.  But this is not the same as the left hand side equals the right hand side%2C which is what needs to be proven.  Could someone explain why the proof works%3F  
1|1|0|||||RE%3A Properties of GCD and LCM%3A Lemma proof|alexb||00:34:31|09/26/2010|John%2C you are right here and the proof needs a fixing. Will do that tomorrow.%0D%0A%0D%0AWhen p is composite%2C it could be factored %28not uniquely%29 into divisors of N_i %28and%2C naturally%2C M%29. For every prime divisor q of p there is at least one i such that N_i is divisible by the highest power of q in lcm. So%2C for every i%2C N_i is associated with the product of maximum powers of q%27s.%0D%0A%0D%0ASomething%27s along these lines.%0D%0A
2|2|1|||||RE%3A Properties of GCD and LCM%3A Lemma proof|alexb||10:08:52|09/26/2010|I have modified http%3A%2F%2Fwww.cut-the-knot.org%2Farithmetic%2FGcdLcmProperties.shtml%23lemma%0D%0A%0D%0AIn fact the only change that was needed was replacing prime p with a power p%5Bsup%5D%26alpha%3B%5B%2Fsup%5D.%0D%0A%0D%0ALCM of several numbers is divisible by p%5Bsup%5D%26alpha%3B%5B%2Fsup%5D iff one of the numbers is. One needs only to pick the number with the largest %26alpha%3B.%0D%0A%0D%0AThank you for pointing out the mistake.