Primes on the GMAT
Think back to your childhood. Do you remember when you would scheme to throw a snowball, grab an extra cookie or give your broccoli to the dog? And do you also remember how, unlike in the movies, your parents would often seem to know what you were about to try to do and thwart your attempt? Sometimes the GMAT is like your parents – it knows what are are thinking even before you do.
One classic example of this is prime numbers. It is not a secret that knowing that a prime number is a positive integer with exactly two factors, itself and one, is essential to the GMAT. However, the test makers also know that some test takers, in an effort to save time, will not always make sure the number being tested is truly prime.
Two numbers, in particular, will show up regularly on the GMAT for this reason. Always be on the lookout for 51 and 91 in any question that deal with primes.
Now take a moment and think about why 51 and 91 are notable. The first idea most students have is that these are both prime numbers. But the truth is trickier than that. 51 and 91 are not prime, but appear to be so at first glance. 3 x 17 = 51 and 7 x 13 = 91. The GMAT knows that most students only learn times tables up to twelve and that the multiplication in each of these cases, therefore, will not be memorized. Thus, 51 and 91 can be especially tricky when they show up in questions that test your knowledge of prime numbers.
Below you will find a sample GMAT problem that deals with this concept. As you try to solve it, keep in mind that you always want to be sure a number is prime before deciding that it is, since some numbers may seem prime, even though they are not.
What is the value of the integer p?
(1) p is a prime number.
(2) 88 ≤ p ≤ 95
Step 1: Analyze the Question Stem
This is a Value question, so we’ll need one exact value for p. There’s nothing to simplify in the question, but it’s worth noting that p is an integer—we won’t need to consider decimal values. So what we need is very clear—one specific numerical value for p.
Step 2: Evaluate the Statements
Statement (1) doesn’t give us one exact value, as there are many prime numbers. Eliminate (A) and (D).
Likewise, Statement (2) doesn’t give us one exact value, only a range with eight possibilities. Eliminate (B).
To evaluate (C) and (E), we must consider these statements in combination. Treating (1) and (2) as one long statement, we know that p is between 88 and 95, inclusive, and that it’s prime. If you happen to have all the primes through100 memorized, then you know right away that p can only equal 89 and that the answer is (C).
But what if you don’t have all those primes memorized?
When evaluating a reasonably short list of numbers, it’s often beneficial to write out the possibilities on your scratch paper. Then, instead of the abstract “88 < p < 95,” we have p = 88, 89, 90, 91, 92, 93, 94, or 95. A prime number is a number that is divisible only by 1 and itself. So, any of these that are divisible by any other number can be scratched off the list.
If we can scratch off seven of these eight numbers, we’ll know p. Any even number is divisible by 2, so that scratches off 88, 90, 92, and 94. Any number that ends in a 0 or a 5 is divisible by 5, so that eliminates 95 (and 90, were it not already gone). We know that 93 is divisible by 3. (The divisibility test for 3 is if the digits of a number add to a multiple of 3, that number is itself divisible by 3. 93’s digits are 9 and 3 . . . 9 + 3 = 12 . . . 12 is a multiple of 3 . . . so 93
is a multiple of 3.)
Now we’re down to p = 89 or 91. We’ve checked for divisibility by the primes 2, 3, and 5. What about the next prime, 7? There is a little-known way to test divisibility by 7. But the GMAT often rewards test takers who think about numbers in creative ways. Because 91 is also 70 + 21, factored out that’s 7(10) + 3(7) or 7(10+3) which brings us to 7(13). So, it’s definitely a multiple of 7 and can be scratched off. We now know the exact value of p . . . p = 89. Together the statements are sufficient. Answer (C).
(Incidentally, that little-known divisibility test for 7 is this: Separate the units digit from the rest of the number, then multiply that units digit by 2. Subtract that from what’s left of the original number. If the result is a multiple of 7, the original number is a multiple of 7. Here’s how that works for 91. Separate 91 into 9 and 1. Multiply 1 by 2: 1 X 2 =
2. Subtract: 9 – 2 = 7. 7 is obviously a multiple of 7, so 91 is a multiple of 7. Try it out on other multiples of 7, and you’ll see that it works every time.)