# 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 2^{2}×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 2^{2}×5 ×7^{2} |
The prime factorization is 2^{3}×5^{2} ×7 |

The greatest common factor will be the smallest number of factors that the
prime factorizations share. 2^{2} ×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 ×3^{2}×5^{2} |
The prime factorization is 2^{2}×3 ×5^{3} |
The prime factorization is 2^{4}×3^{3} |

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 2^{2}×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. 2^{2}×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 2^{2}×5 ×7^{2} |
The prime factorization is 2^{3}×5^{2} ×7 |

The least common multiple will be the largest number of
factors that the prime factorizations share. 2^{3} ×5^{2}×7^{2}=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 2^{2}×7^{2}×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?

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

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