Division: Least Common Multiple
Algebra Help: Section 2.30
Learn to find the least common multiple with algebra.
A Multiple of a quantity is another quantity exactly divisible by it.
Note - If a quantity be multiplied by any factor, the result is properly
called a multiple of that quantity.
This really signifies the meaning of the word multiple
.
But the product is always divisible by the multiplicand;
hence the definition given above.
A Common Multiple of two or more quantities is one quantity which is exactly divisible by each of them.
The Least Common Multiple of two or more quantities is the least quantity exactly divisible by each of them.
It's evident that one quantity, to be divisible by several other quantities, must contain all the factors in each of the given quantities.
1. What is the least common multiple of 8a2x2y,
and 12a3b3x?
OPERATION:
8a2x2y = 23 * a2 * x2 * y 12a3b3x = 3 * 22 * a3 * x * b3. _____________________________________________ 24a3b3x2y = 3 * 23 * a3 * x2 * y * b3
ANALYSIS -
We resolve the given quantities into their component
factors, and write the powers of each factor in the same column. The
different prime factors are 3, 2, a, x, y, and b. The least common
multiple must contain not only each of these factors, but the highest power of
each that is contained in the given quantities, otherwise it will not contain
all the factors of the given quantities. By inspecting the powers of
each, the highest are the 1st power of 3, 3rd power of 2, 3rd power of a,
2nd power of x, 1 power of y, and 3rd power of b.
Their product is 24a3b3x2y, the least common multiple required.
Hence the following rules.
Rules
- Resolve the given quantities into factors, and write all the powers of each factor in the same column.
- Multiply together the higest powers of all the factors, and the product will be the least common multiple sought.
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Page 74-75
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New Elementary Algebra
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By Horation N Robinison, LL. D. Ivison, Blakeman & Company, Publishers, New York and Chicago.
