Greatest Common Factor and Least Common Multiple

Greatest Common Factor

We can use factoring to find the largest number that is a factor of two other numbers.

Find the greatest common factor of 84 and 105.

Find the prime factorization of 84	Find the prime factorization of 105

The prime factorization is 2²×3 ×7	The prime factorization is 3 ×5 ×7

The greatest common factor will be the smallest number of factors that the prime factorizations share. 3 × 7(Think intersection {2, 2, 3, 7} ∩ {3, 5, 7})

Find the greatest common factor of 980 and 1400.

Find the prime factorization of 980	Find the prime factorization of 1400

The prime factorization is 2²×5 ×7²	The prime factorization is 2³×5² ×7

The greatest common factor will be the smallest number of factors that the prime factorizations share. 2² ×5 ×7 (Think intersection {2, 2, 5, 7, 7} ∩ {2, 2, 2, 5, 5, 7})

Find the greatest common factor of 450 and 1500 and 432.

Find the prime factorization of 450	Find the prime factorization of 1500	Find the prime factorization of 432

The prime factorization is 2 ×3²×5²	The prime factorization is 2²×3 ×5³	The prime factorization is 2⁴×3³

The greatest common factor will be the smallest number of factors that the prime factorizations share. 2×3
(Think intersection {2 , 3, 3, 5, 5}∩{2, 2, 3, 5, 5, 5}∩{2, 2, 2, 2, 3, 3, 3})

Another method to find the greatest common factor of 450 and 1500 and 432.

Write the numbers in a row and divide by a prime that will divide all three.

Continue dividing by primes that will divide all three.
There are no primes that will divide all three. The GCF is 6

Least Common Multiple

We can use factoring to find the smallest number that two other numbers will divide.

Find the least common multiple of 84 and 105.

Find the prime factorization of 84	Find the prime factorization of 105

The prime factorization is 2²×3 ×7	The prime factorization is 3 ×5 ×7

The least common multiple will be the largest number of factors that the prime factorizations share. 2²×3 ×5 ×7 = 420 (Think union {2, 2, 3, 7} ∪{3, 5, 7})

Find the least common multiple of 980 and 1400.

Find the prime factorization of 980	Find the prime factorization of 1400

The prime factorization is 2²×5 ×7²	The prime factorization is 2³×5² ×7

The least common multiple will be the largest number of factors that the prime factorizations share. 2³ ×5²×7²=9800 (Think union {2, 2, 5, 7, 7} ∪{2, 2, 2, 5, 5, 7})

Find the least common multiple of 28 and 34 and 98.

Write the numbers in a row and divide by a prime that will divide all three.

If there is no common factor that will divide all three,
find a prime that will divide two of the numbers.

There are no common divisors for the numbers (they are all prime.) The LCM is 2²×7²×17 = 3332

A choral director needs to divide 180 men and 144 women into all-male and all-female
singing groups so that each group has the same number of people. What is the largest
number of people that can be placed in each group?

GCF or LCM ?

Five groups of 36 men and four groups of 36 women.

A movie theater runs its films continuously. One movie is a short documentary that
runs for 40 minutes. The other is a film that runs 100 minutes. Each film runs in a
separate theater. Both movies begin at noon. When will the films begin again at the
same time?

GCF or LCM?

The movies will start together every 200 minutes or every 3 hours and 20 minutes.
They will start together at 3:20 p.m.

Euclid found a way of calculating the GCF without factoring.

Divide the smaller number into the larger, then divide
the remainder into the first divisor. Continue until the
remainder is zero . The last divisor is the GCF.

Find the GCF of 308 and 66

Find the GCF of 168 and 90

Find the GCF of 234 and 470