SOLUTION: Why is the product of two numbers not always their least common multiple?

Question 123106This question is from textbook Introductory Algebra
: Why is the product of two numbers not always their least common multiple? This question is from textbook Introductory Algebra

Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
Because each of the numbers may have a factor in common that will affect the outcome of
finding the least common multiple.
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Let's look at an example. Suppose you are asked to find the least common multiple of 4 and 6.
.
If you multiply those two numbers together you get 4*6 = 24. But 24 is not the least common
multiple. The least common multiple is 12, which is verified by the fact that both 4 and 6
divide evenly into 12.
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The reason for this is that 2 is a common factor of both 4 and 6. 4 is equal to 2*2 and
6 is equal to 2*3. Count the number of factors of 2 in 4 and you have 2. So the answer must
have at least 2 factors of 2. Then look for the number of "new" factors required by the 6.
It has one factor of 2 (it is already accounted for by the 2 factors of 2 needed by
the 4) and one factor of 3 (it is not yet accounted for). So the least common multiple will
be comprised of 2 factors of 2 and 1 factor of 3 ... making it 2*2*3 = 12.
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Hopefully, this gives you an idea of what to look for in trying to find the least common
multiple. Look for the common factors in each of the numbers and if there are common factors
make sure that you have just enough to produce each number. Then multiply those groupings
of factors to get the least common multiple.
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One more example. How about the least common multiple of 12 and 18. If you multiply them
together you get 12*18 = 216.
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But if you look at the factors of 12 you can break 12 down into 2*2*3. And if you look at
the factors of 18 you can break it down to 2*3*3. From this you can see that the most factors
of 2 in one of the numbers is 2*2. And the most factors of 3 in one of the numbers is 3*3.
So the least common multiple of 12 and 18 comes from (2*2)*(3*3) = 4*9 = 36.
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36 is divisible evenly by both 12 and 18 and is the least common multiple of those two
number.
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Think about this a little more, and you'll get the idea.
.
Hope this helps you to understand the concept.
.
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