Prime Factorization, the Greatest Common Factor, and the Least Common Multiple
It is often useful to write a number in terms of its prime factorization, or as the product of its prime factors. For example, 56 can be written as 2×2×2×7 and 84 can be written as 2×2×3×7 . Every number can be written as a product of primes, and, like a fingerprint, every number has a unique prime factorization.
To take a prime factorization of a number, start by dividing the number by its lowest prime factor. Write down this factor, and divide the new number by its lowest prime factor (it does not matter if this is the same as the first prime factor). Write this factor down and divide the new number by its lowest factor. Continue in this manner until the resulting number is prime. Write this number down as the final factor.
Example 1: Compute the prime factorization of 1,575.
Step 1. Is 1,575 divisible by 2? No. By 3? Yes. 1, 575/3 = 525 . Write down 3.
Step 2. Is 525 divisible by 3? Yes. 525/3 = 175 . Write down 3.
Step 3. Is 175 divisible by 3? No. By 5? Yes. 175/5 = 35 . Write down 5.
Step 4. Is 35 divisible by 5? Yes. 35/5 = 7 . Write down 5.
Step 5. 7 is prime. Write down 7.
Therefore, the prime factorization of 1,575 is 3×3×5×5×7 .
Example 2. Compute the prime factorization of 23,100.
Step 1. 23, 100/2 = 11, 550 . Write down 2.
Step 2. 11, 550/2 = 5, 775 . Write down 2.
Step 3. 5, 775/3 = 1, 925 . Write down 3.
Step 4. 1, 925/5 = 385 . Write down 5.
Step 5. 385/5 = 77 . Write down 5.
Step 6. 77/7 = 11 . Write down 7.
Step 7. 11 is prime. Write down 11.
Therefore, the prime factorization of 23,100 is 2×2×3×5×5×7×11 .
Greatest Common Factor
A common factor of two numbers is a factor that divides both numbers. The
greatest common factor (GCF) of two numbers is the greatest number that
divides both numbers. To find the GCF, take the prime factorization of both
numbers. Then write down the factors that they have in common. If they share
more than one of the same factor (two 2's, for example), write them both down.
Then multiply the factors they have in common.
For example, the greatest common factor of 1,575 and 23,100 is 3×5×5×7 = 525 . 1,575 and 23,100 are both divisible by 525, and they are not both divisible by any number greater than 525.
Sometimes, two numbers do not have any prime factors in common. For example, the prime factorization of 40 is 2×2×2×5 and the prime factorization of 21 is 3×7 . Since 40 and 21 have no common prime factors, they are said to be relatively prime, and their greatest common factor is 1.
Least Common Multiple (LCM)
The least common multiple, or LCM, of two numbers is the smallest number
that is divisible by both numbers. To find the LCM, take the prime
factorization of both numbers. Then make a list of the "minimum" factors
required to obtain both numbers. If the prime factorization of one number
contains two 3's and the prime factorization of the other number contains five
3's, write down five 3's.
For example, the least common multiple of 1,575 and 23,100 is 2×2×3×3×5×5×7×11 = 69, 300 . 69,300 is divisible by both 1,575 and 23,100, and there is no number smaller than 69,300 that is divisible by both.
Another way to find the LCM is to multiply the two numbers and divide by the GCF. For example, 1, 575×23, 100 = 36, 382, 500 . 36, 382, 500/525 = 69, 300 . This method is useful when one has a calculator and has already calculated the GCF.
If two numbers are relatively prime, their LCM is the same as their product.
Using the second method for calculating the LCM, it is easy to see why this is
true. The greatest common factor of two relatively prime numbers is 1, so when
the two numbers are multiplied and the result is divided by 1 (the GCF), the
result does not change.
The least common multiple of 21 and 40, since they are relatively prime, is 21×40 = 840 .
Finding the GCF and LCM for Several Numbers
PARGRAPH It is possible to take the GCF or LCM of more than two numbers. To take the GCF, simply multiply the factors that all the numbers have in common. To take the LCM, multiply the minimum factors required to obtain all the numbers (here, you cannot simply multiply all the numbers and divide by the GCF).