How To Write The Prime Factorization In Exponent Form

The fundamental theorem of arithmetic says that each positive integer has a unique factorization. On the surface of it, this seems false. For example, 24 = 2 x 12 and 24 = 6 x 4, which seems like two different factorizations. Though the theorem is valid, it requires that you represent the factors in a standard form – as the exponents of the ordered primes. Prime numbers are those that do not have any proper factors – no factors that are not 1 or the number itself.

Step 1

Factor the number. If any of the factors you find are composite – not prime – continuing factoring until all of the factors are prime. For example, 100 = 4 x 25, but both 4 and 25 are composite, so continue until you get the following result: 100 = 2 x 2 x 5 x 5.

Step 2

Arrange the factors in terms of the primes in ascending order until you have included the largest prime factors in the factor list. For 100 = 2 x 2 x 5 x 5, this would mean 2 (two of these), 3 (none of these), 5 (two of these) and 7 and higher (none of these). For 147 = 3 x 7 x 7, you would have 2 (none of these), 3 (one of these), 5 (none of these), 7 (two of these) and 11 and higher (none of these). The first few primes in order are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.

Step 3

Write the unique factors by writing the exponents only up until the zeros start repeating. So 100 = 2 x 2 x 5 x 5 can be written as 2 0 2 and 147 = 3 x 7 x 7 can be written as 0 1 0 2. Written this way each factorization is unique. To make it easier to read, the unique factorizations are usually written as 100 = 2^2 x 5^2 and 147 = 3 x 7^2.