How Many Prime Numbers Between 1 and 100 are Factors of 7,150? GMAT Problem Solving

Question: How many prime numbers between 1 and 100 are factors of 7,150?

1. One
2. Two
3. Three
4. Four
5. Five

Correct Answer: D

Solution and Explanation
Approach Solution 1:

As per the question, the candidate needs to find out:

How many prime numbers between 1 and 100 are factors of 7,15?
Generally there is no easy way to check whether some very large number is a prime (well if it doesn't have some small primes, which are easy to check). You'll need a computer to do this.
Next, the GMAT usually won't give you a large number to factorize if there is no shortcut for that.

For example in our original question after you find 2, 5, and 7, just check for the next prime 11: \(\frac{143}{11}=13\)

You can also use divisibility rule for 11: if you sum every second digit and then subtract all other digits and the answer is divisible by 11, then the number is divisible by 11. So, for 143: (1+3)-4=0 --> 0 is divisible by 11 thus 143 is divisible by 11.
Or, you can notice that 143=130+13=13*10+13, so 143 must be divisible by 13.
So, as you can see there are plenty of shortcuts to get prime factorization of the numbers from the GMAT problems.

Approach Solution 2:

The speed with which you 'prime-factor' 715 into its 'pieces' is likely going to be influenced by the 'first' number you factor out.
Looking at 7,15, you could easily start with a 2 (because 715 is even), a 5 (because 715 ends in a 0) or a 10 (also since it ends in a 0).
I actually started with 50, since 50 divides into 100 twice.....7100 = (71x2) fifties.…
So 715 = 142 fifties + 1 fifty =
(50)(143)
The (50) can be quickly broken down into (2)(5)(5)
Now, looking at the 143, we know that NO even numbers will divide in (since even numbers do NOT divide into odd numbers). If you know the 'rule of 3', then you know that 3 does NOT divide in. Since 3 doesn't divide in, 9 won't divide in either. won't divide in for obvious reasons. Thus, we're really left with just a handful of possibilities:

1.  143 might be prime
2.  7, 11 and/or 13 might divide in

It's pretty easy to eliminate 7 as an option (it divides into 14, but not 3). Once you find that 11 divides in, you end up with the 13 by default.

Approach Solution 3:
Prime factors of 7150=2×5×5×11×13

It is clear that here are four prime factor that is 2,5,11,13 that are in between 1 and 100

So correct answer will be option D

“How many prime numbers between 1 and 100 are factors of 7,150?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Official Guide Quantitative Review".To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

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