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26 Mar 2009, 06:14
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If we have a huge number, something like 198,921,381,528,567,255, and we want to know if it is divisible by 15, it seems like all we have to do is check to see if the original number is divisible by 5 (which this is) and if the original number is divisible by 3, which this is. Why is this? Why will there not be any situations where the number is divisible by 3 and also by 5 but still not by 15? It makes sense to me that we could first determine it's divisible by 5 and then that result would have to be tested for divisibility by 3, but that's not necessary. Anyone know why?
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26 Mar 2009, 08:15
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If a number is divisible by 3, it has a 3 in its prime factorization. If a number is divisible by 5, it has a 5 in its prime factorization. If x is divisible by 3 and 5, we know that the prime factorization of x looks like 3*5*k, so x is a multiple of 3*5 = 15.
In general, if you know that a number is divisible by a and b, you can always conclude that your number is divisible by the LCM of a and b. So, if you know, say, that x is divisible by 4 and by 6, you can be certain that x is divisible by 12. Don't take the product - you can't be certain that x is divisible by 24.
In general, if you know that a number is divisible by a and b, you can always conclude that your number is divisible by the LCM of a and b. So, if you know, say, that x is divisible by 4 and by 6, you can be certain that x is divisible by 12. Don't take the product - you can't be certain that x is divisible by 24.
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