9514 1404 393
Answer:
- LCM(21, 51) = 357
- LCM(42, 90) = 630
Step-by-step explanation:
For a pair of numbers, I find it convenient to think in terms of factors unique to the number, and factors shared with the other number.
Consider the numbers 21 and 51, for example.
21 = 3·7
51 = 3·17
The factor 3 is shared by both numbers. The factors 7 and 17 are unique to one number or the other.
If we group these factors like this ...
(factors unique to 1 [ shared factors ) factors unique to 2]
= (7 [3 ) 17]
The numbers in ( ) parentheses are the factors of 21, and the numbers in [ ] brackets are the factors of 51.
The LCM (least common multiple) is the product of the factors in those brackets:
LCM(21, 51) = 7 × 3 × 17 = 357
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The unique factors of the numbers don't have to be <em>prime</em> (as in the above example); they just cannot be <em>shared</em>. Here's another example for the numbers 42 and 90
(7 [ 3·2 ) 3·5 ] = (7 [ 6 ) 15] . . . . factors of (42) and [90]
Then the LCM is ...
LCM(42, 90) = 7 × 6 × 15 = 630
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<em>Additional comment</em>
What we have called "shared factors" here is the same as the "greatest common divisor"(GCD) or "greatest common factor" (GCF).
If we divide our little diagram so it shows the product of the two numbers:
(42)[90] = (7×6)·[6×15]
we can see that the LCM is the quotient of the product and the GCD.
LCM(A, B) = A·B/GCD(A, B)
This is an occasionally handy relationship, as it is not difficult to find the GCD of a pair of numbers using Euclid's algorithm.
