Finding GCF + LCM with Exponents

We can also use exponents to find greatest common factors and least common multiples if we write our prime factorizations using exponents.

Using Exponents to Find the GCF

To find the greatest common factor of 50,400 and 2,058,00, we want to compare the exponents on their prime factors and chose the smaller exponent.

50,400 = 25 × 32 × 52 × 71

294,000 = 24 × 31 × 53 × 73

For example, if we look at the exponents on the prime factor 2, we see that 50,400 has five 2’s and 294,000 has four 2’s. The greatest number of 2’s that a common factor can have and still fit inside of both numbers is four. If the greatest common factor had five 2’s, it wouldn’t be a common factor because it wouldn’t fit inside of 294,000; and if it had three 2’s, it wouldn’t be the greatest common factor because we could make it greater by adding another 2.

We know 8,400 is a common factor of 50,400 and 294,000 because its prime factorization fits inside of the prime factorizations of both 50,400 and 294,000. It is the greatest common factor because it is impossible to add another prime factor without going outside of one of those prime factorizations.

Using Exponents to Find the LCM

To find the least common multiple of 50,400 and 2,058,00, we want to compare the exponents on their prime factors and chose the greatest exponent.

50,400 = 25 × 32 × 52 × 71

294,000 = 24 × 31 × 53 × 73

If we look at the exponents on the prime factor 3, we see that 50,400 has two 3’s and 294,000 has one 3. The least number of 3’s that a common multiple can have and still fit both numbers inside of it is two. If the least common multiple had one 3, it wouldn’t be a common multiple because 50,400 wouldn’t fit inside of it; and if it had three 3’s, it wouldn’t be the least common multiple because we could make it smaller by taking away one of the 3’s.

We know 176,400 is a common multiple of 50,400 and 294,000 because the prime factorizations of both 50,400 and 294,000 fit inside of its prime factorization. It is the least common multiple because it is impossible to remove one of the prime factors and still fit both prime factorizations inside.

Practice Using Exponents to Find the GCF or LCM

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