Everyone studying for the GMAT wants to identify the skills that will lead directly to the greatest point increases. While this can be difficult to do, given the adaptive nature of the exam, some skills definitely do come into play more often than others.
One of the most important skills to master for the GMAT is prime factorization. Finding prime factors can be useful on many different types of questions. On test day, if you are stuck on a question and unsure of how to solve, remember the big number rule. The big number rule is simply this: if you see a big number, one that is so large it is unreasonable to work with, find its prime factors. Once you have those factors, you should be able to simplify.
Every positive integer that is not prime, with the exception of 1, can be broken down into a series of prime numbers multiplied together. Additionally, each series of primes is unique and will only result in a single integer.
For example, let’s say you want to find the prime factors of 20. Start by identifying any two numbers that multiply to equal 20. We will choose 4 x 5. 5 is prime, so we cannot break it down further, but 4 is not, so we repeat the process. If you try to think of two numbers that multiply to equal 4, you will only come up with 2 x 2, which are both prime. Thus, the prime factorization of 20 is 2 x 2 x 5.
Now let’s say that instead of choosing 4 x 5, you choose 2 x 10. 2 is prime, so we leave it alone, but we need to break down 10. 10 equals 2 x 5, thus we again find the prime factorization of 20 is 2 x 2 x 5. Notice that no matter what numbers we choose, as long as the math is correct, we reach the prime factors.
Try the question below and see if you can figure out how prime factorization can help you solve.
Problem:
Is q a multiple of 48?
(1) q is a multiple of 6.
(2) q is a multiple of 8
Solution:
The stem asks whether q is a multiple of 48. It’s a yes/no question so you should see if you can answer the question with both a yes and a no. The prime factorization of 48 is 2 x 2 x 2 x 2 x 3, so we need four 2’s and one 3 to know that q is a multiple of 48.
Statement 1: This tells you that q is a multiple of 6. The prime factorization of 6 is 2 x 3, so we know q has one 2 and one 3, but we do not know if it has the additional three 2’s we need, so this statement is not sufficient.
Statement 2: If q is a multiple of 8, we know that it must include the prime factors 2 x 2 x 2, which gives us three 2’s. Since we do not know if it will have the additional 2 and one 3, it is insufficient.
Statements 1 & 2: If q is a multiple of 6 and 8, it must have one 2 and one 3 in its prime factors, so it will divide by 6, and three 3’s in its prime factors, so it will divide by 8. This means that q must have three 2’s and one 3. Note that one of those 2’s is being used to divide by 6 and by 8, since we need to be able to divide by both, but not at the same time. However, to reach 48 the prime factors would need to include one additional 2. Since we do not know if q would include a fourth 2 or not, the statements are still not sufficient. Therefore, the answer is (E).