LCM 120, HCF 4Date: 01/24/2002 at 05:22:32 From: WG Subject: Factorisation Question: Find two numbers if their LCM is 120 and their HCF is 4. (Give three possible answers). I have tried listing all the factors of 120 (e.g. 6x20, 4x30), but none of them fits the description. 1x120 is wrong because 1 is LESS than 4. 2x60 is also wrong because 2 is less than 4 also. Same goes for 3x40. 4x30 is wrong because 30 cannot be divided by 4... and none of the others works either. :( Please help. I would really appreciate it. Date: 01/24/2002 at 09:01:36 From: Doctor Ian Subject: Re: Factorisation Hi, Let's call the two numbers A and B. If the HCF of the two numbers is 4, then the prime factors must be A = 2 * 2 * (the other prime factors of A) B = 2 * 2 * (the other prime factors of B) And the sets of other prime factors can't have any factors in common, because then the HCF would be something other than 4. Does that make sense? So we're looking for two numbers, both multiples of 4, that share no prime factors other than a pair of 2's, and that both divide 120 evenly. How can we find them? Well, let's look at the prime factors of 120: 120 = 2 * 60 = 2 * 2 * 30 = 2 * 2 * 2 * 15 = 2 * 2 * 2 * 3 * 5 Suppose I divide up the factors this way: A = 2 * 2 * 2 = 8 B = 2 * 2 * 3 * 5 = 60 What's the largest number that divides both of these numbers? Looking at the prime factors, we can see that it's 4. So the HCF is 4. What's the smallest number that both of these will divide evenly? It would be 2 * 2 * 2 2 * 2 * 3 * 5 ----------------- 2 * 2 * 2 * 3 * 5 = 120 \___/ | +-- These appear in both numbers, so we only count them once. So 4 and 60 would appear to be one answer to the problem. Can you find some other solutions? Note that your assumption that the LCM of two numbers is the product of the numbers is sometimes correct, but sometimes not. By thinking about prime factors, can you come up with a rule for predicting when LCM(a,b) = a*b will be true? I hope this helps. Write back if you'd like to talk more about this, or anything else. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 01/24/2002 at 09:39:34 From: WG Subject: Factorisation Thank you so much :) I never did dream that you would reply so soon! I have an ambition to be a mathematician or cosmologist when I 'grow up'. If I do become one I'll remember (try to anyway) your homework help service :) Thanks again! |
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