MATH 035
Penn State University
Dr. James Sellers
Handout: Least Common Multiples
Let a and b be positive integers.
Definition: We say that the least common multiple of a and b is the smallest number that is a
multiple of both a and b. We denote this number by lcm(a,b).
Examples:
lcm(4, 6) = 12
lcm(10, 55) = 110
lcm(32, 80) = 160
lcm(35, 36) = 1260
Connection to GCDs:
Notice that GCDs and LCMs play a somewhat “opposite role” to one another. On the one hand,
a GCD is the largest divisor of two given numbers, while the LCM is the smallest multiple of the
same two numbers. I like to think of this relationship with the following diagram:
lcm(a, b)
ր
տ
a
b
տ
ր
gcd(a, b)
Techniques for Calculating LCMs:
One “recipe-like” way to find the least common multiple of two numbers a and b is to do the
following:
1) Find the prime factorizations of each of the two numbers. For notation’s sake, let’s write
them this way:
a = p1c1 p2 c2 ⋯ pr cr
b = p1d1 p2 d2 ⋯ pr dr
2) Choose the powers of each prime that are larger and multiply those together. That product
is the least common multiple of the two numbers with which you started!
Examples:
A) Let a = 100 and b = 240. The two prime factorizations are as follows:
a = 2 2 × 52
b = 24 × 3 × 5
In order to get the same primes to appear in each of the two factorizations, we rewrite
them slightly as follows:
a = 2 2 × 30 × 52
b = 24 × 31 × 51
Then we see that lcm(100, 240) = 24 × 31 × 52 = 1200.
B) Let a = 242 and b = 77. The two prime factorizations are as follows:
a = 2 ×112
b = 7 × 11
In order to get the same primes to appear in each of the two factorizations, we rewrite
them slightly as follows:
a = 21 × 7 0 × 112
b = 20 × 71 ×111
Then we see that lcm(242, 77) = 21 × 71 × 112 = 1694.
Exercises:
Using prime factorizations, determine the following:
lcm(182,26) = __________________________________________
lcm(950,100) = _________________________________________
lcm(81,144) = __________________________________________
lcm(770,231) = _________________________________________
lcm(4250,25992) = ______________________________________
lcm(165528,203148) = ___________________________________
lcm(248292,111078) = ___________________________________
lcm(71415,352843) = ____________________________________
Connecting GCDs to LCMs:
Given the calculations you made above, and the GCD calculations you completed on the same
pairs of numbers above in a past handout, can you find a connection between GCDs and LCMs?
Given that connection, here’s a second way to calculate LCMs!!
© 2010, James A. Sellers