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Prime Factorization

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.

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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.

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Relative Primes

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.