How can I describe an angle's measure?
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It's a factor. This concept is widely used throughout algebra, and you'll probably bump into it through the end of high school and beyond.
A common use is expressing a term in <em>prime factorization</em>, or reducing a number to its most base parts- primes. For example:
Of course, a number like 13 which is already prime is made up of itself and 1. <em>Factors do not have to be primes.</em> 20 is also reducible through combinations of 1, 2, 4, 5, 10, and 20. Prime factorization is just a handy example.
Basically, factors multiply with each other to create other numbers, and numbers can be reduced down to their factors.