How to Methodically Crack Tough “Primeness” Questions on the GMAT

Questions that ask you to determine if a number is prime are ubiquitous on the GMAT. You can expect to come across at least a few on exam day, so knowing how to quickly determine a number’s “primeness” will be necessary if you’re looking to break the 700 ceiling.

For most 600-700 difficulty questions, memorizing all prime numbers from 2-101 (remember, 1 is NOT considered prime) is probably easiest. Occasionally, however, you may find yourself facing a particularly large integer – this is where the above tactic comes up short.

In these cases, one strategy I recommend also relies on memorization, but it has the added benefit of incorporating divisibility rules (sort of a ‘kill two birds’, approach to speed up your studying). By applying divisibility rules to large integers, you can create a checklist that will build confidence in your assessment of “primeness”.

Example

Consider the following: “Is integer X a prime number?” 1) X is not divisible by 3. 2) X = 143.

Ignore, for a moment, that with statement 2, we can easily mark B and move on (since if we know the integer, we can determine its “primeness”). As a mental exercise, think about how you might assess the question.

At first blush, we can immediately cross out A and C, since statement 1 is clearly insufficient. Statement 2, while a bit trickier, is still very doable in a reasonable amount of time:

• 2 = 143 is not even à MOVE ON
• 3 = 1+4+3=7 which is not divisible by 3 à MOVE ON
• 4 = 143 is not divisible by 2, so it can’t be divisible by 4 à MOVE ON
• 5 = Doesn’t end in 5 or 0 à MOVE ON
• 6 = 143 is not divisible by 3 nor 2 à MOVE ON
• 7 = Drop the 3, multiply by 2, subtract from the remaining digits = 8, which is not divisible by 7 à MOVE ON
• 8 = Apply the same logic to 8 as we did to 4 à MOVE ON
• 9 = 143 is not divisible by 3, so it can’t be divisible by 9 à MOVE ON
• 10 = 143 is not divisible by 5 nor 2 à MOVE ON
• 11 = This one is trickier – normally I save 11 as the final number on my checklist. That said: (1-4) + 3 = 0, 0 is divisible by 11, so 143 is divisible by 11, so it cannot be prime à Thus, statement 2 is sufficient.
• 13 = Drop the 3, multiply by 9, subtract from the remaining digits = (-13), which IS divisible by 13, so 143 is divisible by 13, so it cannot be prime à Thus, statement 2 is sufficient.

Looking at the above steps, many of you are probably dubious regarding the efficiency of this approach. Admittedly, the steps make this approach look longer than it is!

The truth is, most of you probably didn’t need to “check off” the rules for 2,4,5,6,8 or 10 (and testing the “factorhood” of 9 automatically tests for 3), which just leaves 7, 11, and 13. Many of you can probably cross off the easier checks in ~20-30 seconds, which leaves you plenty of time to test 7,11, and 13.

GMAT testmakers like to throw these multiples into integers to try to trap testtakers to (falsely) assume a number is prime, since their divisibility rules are less intuitive. Further, while the example I used was nice and small, because any integer is fair game, (e.g., 3,255) knowing these rules can help you in a pinch.

I recommend any testtaker looking to crack the 700 ceiling add 7, 11, and 13 to their running list of divisibility rules, and consider the ‘checklist’ strategy when confronted with pernicious “primeness” problems.

Reference Tool

For reference, here are the rules for all the integer divisibility rules used above:

• 2 = If the integer is even, it is divisible by 2.
• 3 = Add up the digits and if the sum is divisible by 3, then the original integer is divisible by 3.
• 4 = Try to divide the integer by 2 twice.
• 5 = If the last digit ends in a 5 or 0.
• 6 = If the integer is divisible by both 3 and 2.
• 7 = Drop the last digit, multiply by 2, subtract the resulting product from the remaining digits. If the final integer is divisible by 7, the original integer is divisible by 7. Occasionally, you may not be able to immediately see if the resulting quotient is divisible by 7. In cases like these, keep applying the rule.
• E.g., 112 à 2*2 = 4 à 11 – 4 =7 à 112 is divisible by 7.
• E.g., 3,255 à 2*5 = 10 à 325 – 10 = 315 à 2*5 = 10 à 31 – 10 = 21 à 3,255 is divisible by 7.
• 8 = Try to divide the number by 2 three times.
• 9 = Add up the digits and if the sum is divisible by 9, then the original integer is divisible by 9.
• 10 = If the last digit 0, it is divisible by 10.
• 11 = Moving from left to right, alternate between subtracting and adding each digit. If the final integer is by 11, the original integer is divisible by 11.
• E.g., 14,641 à 1-4+6-4+1 = 0 à 14,641 is divisible by 11.
• E.g., 2,816 à 2-8+1-6 = -11 à 2,816 is divisible by 11.
• 13 = Apply the same approach as 7, but use 9 instead of 2 as your multiplier.
• E.g., 156 à 9*6 = 54 à 15 – 54 = -39 à 39÷13 = 3, so 156 is divisible by 13.
• E.g., 1,014 à 9*4 = 36 à 101 – 36 = 65 à 9*5 = 45 à 6-45 = -39 à -39÷13 = -3, so 1,014 is divisible by 13.

The GMAT is a test you can prepare for. It is not an intelligence test. With the GMAT, more than with any other standardized test, having a clear and targeted strategy for your preparation will help you get a high score, while also allowing you to juggle your other priorities. If you’re applying to business school, you probably have a lot on your plate, and need your test preparation to be effective and efficient.

You might be wondering why we’re so confident the GMAT can be mastered. The reason is simple: Because question types recur frequently, knowing instantly which methodologies to apply to each type of question dramatically improves scores. In essence, it allows you to know at a glance what the test is looking for. Your success on the GMAT depends on your ability to:

• Master content and recognize question types
• Strategically leverage the best materials for your specific needs
• Adopt a data-driven approach to diagnostic assessment

Our tutors are experts at guiding students through these challenges. We also understand the make-or-break importance of the GMAT and we're dedicated to helping you beat the exam. Before you even meet with your tutor, we assess your strengths and weaknesses on a diagnostic exam. Based on your results, your tutor will construct a customized syllabus in advance of your first session.

Taking the GMAT this year? Check out some other helpful blog posts below!

GMAT Tip: Look for Dangling Modifiers!

The GMAT was designed with you in mind

That Fine Line between Pure Algebra and Testing Numbers on the GMAT