GMAT Math Review - Factors and Prime Factorization
Factors
A factor of an integer is an integer that divides into that number without leaving a remainder. The integer 3, for example, is a factor of 6, because 3 can divide into 6 giving a quotient (result) of 2 with no remainder. There is an inherent relationship between the concept of factors and the concept of multiples. Just as 3 is a factor of 6, 6 can be said to be a multiple of 3.
Understanding and being able to work fluidly with factors will be important on the GMAT Quantitative Section. A few questions might test your knowledge of the definition and properties of factors. More often, however, you will need to use factoring to solve more complex problems.
All integers have a finite set of factors. A straightforward technique for quickly listing all of the factors of a number is to build a list or table of factor pairs. A factor pair is simply the two factors of a number which when multiplied together produce the number. For example 9 and 5 are a factor pair of 45, as are 15 and 3, and 45 and 1. To systematically build a table of factor pairs, begin with 1 and work your way through all possible pairs until numbers start repeating. As an example, consider the factorization of 48:
| 1 | 48 |
| 2 | 24 |
| 3 | 16 |
| 4 | 12 |
| 6 | 8 |
Begining with 1 matched against the number 48 itself, you can test each successive integer. 48 divided by 2 is 24, 48 divided by 3 is 16, 48 divided by 4 is 12, 5 does not divide evenly into 48, 48 divided by 6 is 8, and 7 does not divide evenly into 48. Now that you've reached 8, the numbers will begin repeating and you know that you have identified all the factors of 48.
Prime Numbers
Prime numbers are integers that have only two factors: 1 and the number itself. Examples of prime numbers are 2, 3, 5 and 7. 4 is not prime because it can be divided by 2, and 6 is not prime because it can be divided by 2 and 3. It is important to remember that 1 is not considered a prime number because it does not fit the definition of only two factors. Most GMAT test takers will find it useful to simply memorize the prime numbers to 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.
Prime Factorization and Factor Trees
Prime factorization is the process of determining the prime numbers that when multiplied together equal a number. Note that the prime factorization of a number is not simply the list of prime numbers that are factors. More often than not, specific prime numbers will be repeated several times in a prime factorization. For example, the prime factorization of 18 is $2 * 3 * 3$. When prime numbers are repeated, they are usually expressed as powers, so the prime factorization of 18 would be more succinctly written as $2 * 3^2$.
A common technique for prime factorization is to build a factor tree. As depicted in the diagram below, begin with the number being factored, and take any factor pair make the first two branches of the tree. If either number in the pair is prime, that branch ends (it can be helpful to circle the prime number). If either or both numbers in the pair are not prime, then take a factor pair from that number and make two more branches. In short order, all of your branches will terminate and you will have the prime factorization.
As you can see, using the factor tree we were able to determine the prime factorization of 150 to be $2 * 3 * 5^2$ in just a few steps. Remember that when using this technique it does not matter what factor pair you begin with, the result will always be the prime factorization.
Greatest Common Factor and Least Common Multiple
The greatest common factor (GCF) of two numbers is the largest number that will divide two numbers evenly. For example the GCF of 27 and 18 is 9 because 9 is the largest number that will divide into both 27 and 18 without leaving a remainder.
To find the GCF of two numbers, take the prime factorization of both numbers and then multiply the common factors.
As an example will find the GCF of 98 and 154.
Prime factorization of 98: $2 * 7^2$
Prime factorization of 154: $2 * 7 * 11
Multiply common factors using the lowest powers: $2 * 7 = 14$
When common factors are raised to different powers, choose the lowest power. Returning to our example of 18 and 27, the prime factorization of 18 is $2 * 3^2$ and the prime factorization of 27 is $3^3$. The GCF is $3^2$ (or 9).
Least Common Multiple
The least common multiple (LCM) of two numbers is the smallest number that is divisible by both of the original numbers. For example, 21 is the LCM of 3 and 7 because it is the smallest number that is a multiple of both 3 and 7. You are likely familiar with the concept of the LCM from working with fractions when it is sometimes necessary to find the least common denominator before adding or subtracting fractions.
In the example of 3 and 7, we could have arrived at the LCM by simply multiplying the two numbers together. There will be times however when multiplying the two original numbers will result in a product larger than the LCM. Multiplying 9 and 12 produces 108, but the actual LCM of these two numbers is 36. In a method very similar to the GCF method described above, prime factorization can be used to quickly determine the LCM of any two numbers. Starting with the prime factorization of the number, multiply all of the factors together using the highest power of any factor the two have in common. Carrying on with the example of 9 and 12:
Prime factorization of 9: $3^2$
Prime factorization of 12: $2^2 * 3$
Multiply common factors using the highest powers: $2^2 * 3^2 = 36$
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