Least Common MultipleDate: 10/26/1999 at 13:56:34 From: Michelle Boyce Subject: Common Numbers What is the lowest number that is divisible by the numbers 1 through 10? Someone told me that if I take away 1, 2, 3, 4, 5, 6 and multiply 7 x 8 x 9 x 10 = 5040. I don't understand how they figured that out and what the term for that kind of math is called. Can you help me? I need someone to explain the steps to me. Date: 10/26/1999 at 16:15:41 From: Doctor Rob Subject: Re: Common Numbers Thanks for writing to Ask Dr. Math, Michelle. You are asking for the Least Common Multiple of 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10. Your "somebody" is not correct. The number 5040 is a multiple of all of them, but it is not the least such. To solve such problems, you can write the prime-power factorization of each of the numbers, and the exponent of the prime powers that occur: exponent of number factorization 2 3 5 7 ------ ------------- - - - - 1 1 0 0 0 0 2 2^1 1 0 0 0 3 3^1 0 1 0 0 4 2^2 2 0 0 0 5 5^1 0 0 1 0 6 2^1*3^1 1 1 0 0 7 7^1 0 0 0 1 8 2^3 3 0 0 0 9 3^2 0 2 0 0 10 2^1*5^1 1 0 1 0 Now take the maximum in each column, and use those as exponents for your number: 3 2 1 1 2^3 * 3^2 * 5^1 * 7^1 Now multiply this number out, and you'll have your answer. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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