Take a crack at this problem:
How many positive integers less than 2 x 10^4 are there in which each digit is a prime number?
Just as problems on speeding trains can invoke feelings of dread bordering on public speaking, so too can the previous type of problem, which I refer to as counting properties. Typically, they show up about once on the GRE, and usually elicit a mixture of fear and disgust.
Yet, we shouldn’t let these problems bring us down. To attack them, I use a method I like to call the dash method. To understand how to use the dash method, let’s first try an easier GRE Math problem.
How many three-digit numbers have a prime number in hundreds place and an even number in the tens place?
First off, the tens digit refers to the number in the middle, and the hundreds digit refers to the leftmost number. For example, in the number 245, the 2 represents the hundreds, the 4 represents the tens and the 5 represents the units.
In order to attack this problem, let’s use a dash to stand for the hundreds, tens, and units digit, respectively.
_____ _____ _____
According to the problem, we need a number that is a prime number to go in the hundreds place, or the first of the three dashes above. So how many prime numbers match this requirement?
Most students would immediately say the number 1. But we must remember that a prime is only divisible by itself and 1. 1 is itself, so 1 is not a prime number. The lowest prime is 2. Next, we have 3, 5, and 7. What about 11? Well, in the above problem the hundreds digit can only be one number. If we were to put 11 in the hundreds column, we would then have a four-digit number.
Next, we have to take the total number of possible digits that go in the first dash. We have 2, 3, 5, and 7, so that gives us a total of 4 different numbers. Write that 4 into the leftmost dash. Now, let’s take a look at the middle number, the tens digit. Here we need an even number. That gives us 0, 2, 4, 6, and 8 (yep, zero is even). This is a total of five possibilities. So, write a 5 in the middle dash. Finally, we have the units digit. There are no restrictions here. That is, the numbers 0 – 9 can fit, giving us a total of 10 possibilities.
Using the dash method we get:
4 5 10
What do we do with the dashes? We simply multiply them and there’s our answer.
4 x 5 x 10 = 200.
Now, see if you can try the original problem.
Did you try it out? Here’s the explanation:
How many digits less than 2 x 10^4 are there in which each digit is a prime number?
When using the dash method, we want to use a dash to stand for each of the digits in a problem. In this problem, we are apparently dealing with 5-digit numbers, because 2 x is equal to 20,000. The question asks us to find numbers less than 20,000 where each digit is represented by a prime number. So, let’s first set out our dashes:
___ ___ ___ ___ ___
Above each dash represents one of the five digits, from right to left, of a five-digit number. The question asks us to find how many five digit numbers are composed only of primes.
The first trick to this problem is to notice that the very first digit, the one in the leftmost dash representing the ten-thousands digit, has to be a prime. We know that the primes are 2, 3, 5, and 7. Above, the question states that the number has to be less than 20,000. The highest such number is 19,999.
Do you notice anything fishy? The ten thousands digit can only be a 1. One, however, is not a prime. Therefore, there are no five-digit numbers less than 20,000 in which each individual digit is a prime number.
Does that means we are finished with the problem? What about four-digit numbers? The problem simply says the number has to be less than 20,000, so four-digit numbers are definitely in. So, how many are composed of only prime numbers?
Now, we can use the dash method as our go-to math trick. Remember, over each dash we want to place the total number of possibilities. For each of the four digits, there are four different prime numbers—2 , 3, 5, and 7—that could work. Therefore, we will place a 4 over each dash:
4 4 4 4
Now we just have to multiply the total number of possibilities, which is equal to multiply each of the dashes together, giving us 4 x 4 x 4 x 4 = 256.
Therefore, there are 256 four-digit numbers in which every number is a prime (as an example 3, 257 is one such number, as is 2553). Are we finished? Well, what about three-digit numbers? Again we use the dashing method and we get 4 x 4 x 4 = 64.
We are still not done, because now we have to consider two-digit numbers. When we do so we get 4 x 4 = 16.
Finally, we have to consider single digits, which would be 2, 3, 5, and 7. Here, four possibilities go in the one and only dash (no need to multiply dashes), giving us 4.
The last thing we have to do is add together the total instances for four-digit, three-digit, two-digit and one-digit numbers. Doing so gives us 256 + 64 + 16 + 4 = 340, Answer Choice (C).
GRE Math Trick Takeaway
That was a pretty tough problem, but if you are able to follow the steps and apply them to similar problems, you should no longer be cowed by these intimidating GRE math problems.