Sometimes it is necessary or helpful to factor an integer completely. This means you need to find all the factors of that integer. It’s possible that the test will directly require this skill or will make use of it in a more complicated question. In either case, it’s something you should know how to do.

 

 

    To find all the factors of a number, write them down in pairs, beginning with 1 and the number you’re factoring. We’ll factor 24 in this example. One and 24 are both factors of 24. Next, try every integer greater than 1 in increasing order. Here are the factor pairs we find for 24:

• 1 and 24 (124 = 24)
• 2 and 12 (212 = 24)
• 3 and 8 (38 = 24)
• 4 and 6 (46 = 24)

    You know you’ve found all the factors of a number when the next first factor exceeds its corresponding second factor. For example, after you found that 4 was a factor of 24 and 5 was not, you would see that 6, the next factor of 24, had already been included in a pair of factors. Thus, all the factors have been found.

 

 

    A prime number is a number whose only factors are 1 and itself. All prime numbers are positive (because every negative number has –1 as a factor in addition to 1 and itself). Furthermore, all prime numbers besides 2 are odd. The first few primes, in increasing order, are:

 

 

    To determine whether a number is prime, you shouldn’t check whether the number is divisible by every number less than itself. Such an effort would take an incredible amount of time, and you have only an hour for the Math IC. Instead, to decide whether a number is prime, all you need to do is estimate the square root of the number, then check all the prime numbers that fall below your estimate. For example, to see if 91 is prime, you should estimate the square root of the number: . Now you should test 91 for divisibility by the prime numbers smaller than 10: 2, 3, 5 and 7.

• Is 91 divisible by 2? No, it does not end with an even number.
• Is 91 divisible by 3? No, 9 + 1 = 10, and 10 is not divisible by 3.
• Is 91 divisible by 5? No, 91 does not end with 0 or 5.
• Is 91 divisible by 7? Yes! 917 = 13.

    Therefore, 91 is not prime.

 

 

    Another form of factorization is called prime factorization. The prime factorization of an integer is the listing of the prime numbers whose product is that number.

 

    To find the prime factorization of a number, divide it and all its factors until every remaining integer is prime. This group of prime numbers is the prime factorization of the original integer. As an example, let’s find the prime factorization of 36.