## Improve your Mental Math

Fact: On the GMAT, you do not get a calculator.

Actually, on the new IR questions coming this summer, there will be an onscreen calculator, but for the ordinary Quantitative questions, for the foreseeable future, you will be doing those without a calculator of any sort.

Consider the following question:

S = {71, 73, 75, 77, 79, 81, 83, 85, 87, 89}

How many numbers in S are prime?

This, with little variation, could be a Problem Solving question, or it could play a role in some aspect of a Data Sufficiency question. How to determine this without a calculator?

## The Easy Rules

First of all, it’s probably obvious that all even numbers are divisible by 2. Furthermore, 2 is the only even prime number: all other positive even integers are divisible by 2, so all other prime numbers are odd.

A number that ends with a 5 or a 0 is divisible by 5. A two digit number in which the first and last digits are the same is divisible by 11.

With 4, there’s an interesting pattern. Think of the one-digit numbers that are multiples of 4: those are 0, 4, and 8. If the last digit is 0, 4, or 8, and the tens digit is *even*, then the number is divisible by 4. If the last digit is even but not one of those — viz. 2 or 6, the one-digit even numbers that are *not* multiples of 4 – and the tens digit is *odd*, then the number is divisible by 4.

## The Pattern with 3

The rule for 3 is a bit different from the rules for other numbers. You simply add up the digits. If the sum of the digits is divisible by 3, then the original number is divisible by 3. For example, consider the number 285 —- 2 + 8 + 5 = 15, which is divisible by 3, so that means 285 must be divisible by 3. Consider the number 2012 —- 2 + 0 + 1 + 2 = 5, which is not divisible by 3, so that means 2012 is not divisible by 3.

That pattern also works with 9 — if the sum of the digits is divisible by 9, then the original number is divisible by 9. For example, the next year that will be a multiple of 9 is 2016, because 2 + 0 + 1 + 6 = 9.

## Testing Prime Numbers

The GMAT occasionally will have questions that require you to figure out whether larger two-digit numbers are prime. It will not ask you to figure out whether numbers greater than 100 are prime. For numbers less than 100, two-digit numbers, it’s enough to check whether it divisible by any single digit prime number {2, 3, 5, 7}. Any two-digit number not divisible by one of those four must be prime.

Back to our set above:

S = {71, 73, 75, 77, 79, 81, 83, 85, 87, 89}

All odds, so no multiples of 2.The numbers 75 and 85 are obviously divisible by 5, so those are not prime.

S = {71, 73, __, 77, 79, 81, 83, __, 87, 89}

The number 77 = 7*11. Once we have that multiple of 7, we add and subtract 7 to find other multiples of 7: { . . . . 70, 77, 84, 91, . . . } We see that 77 is the only multiple of 7 on this list.

S = {71, 73, __, __, 79, 81, 83, __, 87, 89}

Now we need a multiple of 3. We notice that, with 81, 8+1=9, which is divisible by 3, so 81 must be divisible by 3. We add and subtract 3 repeatedly to get other multiples of three: { . . . 69, 72, 75, 78, 81, 84, 87, 90, . . .}. Just to double-check 87, we see 8+7=15, which is divisible by 3, so of course 87 is divisible by 3. We can eliminate

S = {71, 73, __, __, 79, __, 83, __, __, 89}

The remaining five numbers must be prime. Indeed, each one is a prime number.

Here’s a free GMAT Problem Solving question on which you can practice these skills: http://gmat.magoosh.com/questions/850

That question will be followed by a full video explanation when you submit your answer. Good luck!

How to find the divisibility of 7?

Yes there is! Actually, there is a method for testing divisibility of 7 on the GRE, although it’s a bit more… convoluted than other divisibility tests. Here are the steps:

1) Take the final digit of the number and remove it completely from the number.

EX:203 >>>> 202) Multiply the removed final digit by 2.

EX:3*2 = 63) Subtract that doubled final digit from the number you created in step 1 when you removed the final digit.

EX:20-(2*3) = 20-6 = 14.4) Divide the result of step 3 by 7. If it divides evenly by 7, the whole number is divisible by 7.

EX:14 from step 3 above is divisible by 7, since 14/7 = 2. Thus, 203 is divisible by 7. Tested on a calculator, 203/7 = 29.This method works best for 3 digit numbers. For 2 digit numbers, it’s far easier to just use mental meth to assess divisibility by 7. And for numbers of 4 digits or more, you may need to apply this divisibility test multiple times, so that you can finally get a two digit result to test divisibility by 7 on. Fortunately, you don’t need to worry about 4+ digit numbers for divisibility by 7 on the gRE. The GRE is very unlikely to give you a problem that relies on testing 7-divisibility for a number greater than 1,000.

Hi.. Just to sum up.. to check for 2 digit the divisblity we have to consider only 2, 3 5, 7

Hi Aliya,

Yes, a two-digit number that is not divisible by 2, 3, 5 or 7 will be prime. Happy studying! 🙂

Thanks Mike!

The concept of 2 digit numbers being prime only when they are not divisible by any of {2,3,5,7} is really helpful in GRE questions as well 🙂

Dear Karan,

You are quite welcome! I am glad you found this helpful! Best of luck to you, my friend!

Mike 🙂

Ha ha… I was just surfing around and took a glimpse at these comments. I can’t believe there’s still this much attention. Thanks for writing about this.

Why, thank you. Best of luck to you.

Mike 🙂