ALGEBRA
Sec. 05
MathHands.com
Márquez
LCM INTRO
a working UNDERSTANDING of LEAST COMMON MULTIPLE
We usually call a number m a multiple of some other number b if b times some integer is equal to m.
For example, we call 12 a multiple of 3 since 3 times 4 is equal to 12. In fact, 3, 6, 9, 12, 15, etc... are all
multiples of 3. A layman understanding of the least common multiple concept for two numbers would
be to established by simply calculating one. For example, to calculate the lcm(12, 9) we would first list
several multiples of 12, and several multiples of 9. We then identify the common multiples and choose
the smallest one:
multiples of
multiples of
12 : 12, 24, 36, 48, 60, 72, . . .
9 : 9, 18, 27, 36, 45, 54, 63, 72, . . .
Once we have exposed the common multiples we simply select the smallest one, 36. This should be seen
as a layman perspective on least common multiples. It works well for numbers and it helps ground the
idea. Yet, it has shortcomings. It may not probe as helpful when we find least common multiples of
polynomials. Thus we offer the definitive definition of LCM of two or more terms.
THE DEFINITION of LEAST COMMON MULTIPLE
The definition of the least common multiple is based the prime factorization of the numbers. Once each
of the numbers, a and b is prime factorized, we define the lcm(a, b) as the product of all primes, include
each prime using the larger of the multiplicities where applicable. We take the opportunity to redo the
example above using this definition.
Example: lcm(9, 12)
9 = 3·3
12 = 2 · 2 · 3
TT
BI
Once prime factorized, guts exposed, we can see what each number is made up of. These numbers are
made up of 3’s and 2’s. The 3’s occur once in 12 and twice in 9, thus we will include it twice in lcm. By
definition, we include it the larger of the multiplicities when applicable. The 2’s occur twice in 12 and
none in 9, thus we include it twice in the lcm. Thus the lcm of 9 and 12 is
lcm(9, 12) = 22 · 32 = 36
EXAMPLE of LEAST COMMON MULTIPLE
Find lcm(36x5 y 3 , 90y 7 x4 z 2 )
SOLUTION:
We first prime factorize each term, to get
36x5 y 3 = 22 · 32 · x5 · y 3
90y 7 x4 z 2 = 2 · 32 · 5 · y 7 · x4 · z 2
BI
BI
Then we list all prime factors, 2, 3, x, y, 5, and z. We then decide the multiplicity on each one: 2 appears
twice on the first number, and once on the second one, thus we will include it twice, 3 occurs twice on
each, so we will include it twice, five occurs once on the second, none on the first, so we will include it
once, x will be included 5 times, 7 y’s will be included, and two z’s. Thus
lcm(36x5 y 3 , 90y 7 x4 z 2 ) = 22 · 32 · 5 · x5 · y 7 · z 2
c
2007
MathHands.com
ALGEBRA
Sec. 05
MathHands.com
Márquez
1. List all divisors, identify the common divisors, then find the indicated lcm.
lcm(24, 16)
Solution: lcm(24, 16) = 48
2. List all divisors, identify the common divisors, then find the indicated lcm.
lcm(14, 16)
Solution: lcm(14, 16) = 112
3. List all divisors, identify the common divisors, then find the indicated lcm.
lcm(24, 36)
Solution: lcm(24, 36) = 72
4. List all divisors, identify the common divisors, then find the indicated lcm.
lcm(24, 34)
Solution: lcm(24, 34) = 408
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2007
MathHands.com
ALGEBRA
Sec. 05
MathHands.com
Márquez
5. List all divisors, identify the common divisors, then find the indicated lcm.
lcm(100, 150)
Solution: lcm(100, 150) = 300
6. List all divisors, identify the common divisors, then find the indicated lcm.
lcm(15, 50)
Solution: lcm(15, 50) = 150
7. List all divisors, identify the common divisors, then find the indicated lcm.
lcm(25, 50)
Solution: lcm(25, 50) = 50
8. List all divisors, identify the common divisors, then find the indicated lcm.
lcm(25, 30)
Solution: lcm(25, 30) = 150
9. List all divisors, identify the common divisors, then find the indicated lcm.
lcm(20, 30)
Solution: lcm(20, 30) = 60
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2007
MathHands.com
ALGEBRA
Sec. 05
MathHands.com
Márquez
10. List all divisors, identify the common divisors, then find the indicated lcm.
lcm(60, 90)
Solution: lcm(60, 90) = 180
11. List all divisors, identify the common divisors, then find the indicated lcm.
lcm(65, 25)
Solution: lcm(65, 25) = 325
12. Find the lcm(65, 25, 15)
Solution:
the lcm(65, 25, 15) = 975
13. Find the lcm(45, 9, 15)
Solution:
the lcm(45, 9, 15) = 45
14. Find the lcm(45, 27, 15)
Solution:
the lcm(45, 27, 15) = 135
15. Find the gcd(45, 27, 15)
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2007
MathHands.com
ALGEBRA
Sec. 05
MathHands.com
Márquez
Solution:
the gcd(45, 27, 15) = 3
16. Find the gcd(12, 27, 15)
Solution:
the gcd(12, 27, 15) = 3
17. Prime factorize each of the terms, then find the indicated lcm.
lcm(75x5 , 100x3 )
Solution: 300x5
18. Prime factorize each of the terms, then find the indicated lcm.
lcm(75x5 y 5 , 100x3 y 8 )
Solution: 300x5 y 8
19. Prime factorize each of the terms, then find the indicated lcm.
lcm(75x5 y 5 , 100t3 y 8 )
Solution: 300x5 y 8 t3
20. Prime factorize each of the terms, then find the indicated lcm.
lcm(x5 y 5 t9 , t3 y 8 )
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2007
MathHands.com
ALGEBRA
Sec. 05
MathHands.com
Márquez
Solution: x5 y 8 t9
21. Find the lcm(x, 2x), where x is a natural number.
Solution: 2x
22. Find the lcm(x, x2 ), where x is a natural number.
Solution: x2
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2007
MathHands.com