Least Common Multiple

Date: 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/