CAT Question

[jmuduke08]

CAT 2 Math Q33:

What is the sum of the digits of the positive integer n where n < 99?

1) n is divisible by the square of the prime number y.

2) y4 is a two-digit odd integer.

I understand once the answer told me that the only two digit odd integer for y^4 is 81 but how on earth does one think of that on the test in a reasonable amount of time?

[jnelson0612]

Good question! Let's look at the best way to get through this quickly.

This would be my process with this problem:
"What is the sum of the digits of the positive integer n where n < 99?

1) n is divisible by the square of the prime number y.

2) y4 is a two-digit odd integer."

First, after realizing that there is no good way to rephrase this question to make the question easier, I would evaluate statement 2, which tells me nothing about n. Statement 2 is insufficient. Using the BD/ACE grid, I eliminate BD.

Second, I would evaluate statement 1:
"1) n is divisible by the square of the prime number y."
I would make up some possibilities:
If y is 2, n could be 4, 8, 12, any multiple of 4. This clearly is not going to give me the sum of the digits of n, since each of these three give a different sum. Insufficient. I eliminate A from my grid.

Third, I would evaluate them together. Okay, y^4 must be a two digit odd integer, and y must also be prime. Let's consider what y could be using the primes starting with 2:
Could y be 2? NO, it is prime but 2^4 is not odd
Could y be 3? YES, because 3^4 is 81, a two digit odd number
Could y be 5? NO, because 5^4 is too big. I need y^4 to be a two digit odd number.

Everything after 5 will be even bigger when taken to the fourth power, so y must be 3.

So if y is 3, then n must be divisible by the square of y, or 9. What are numbers divisible by 9 that are less than 99? Multiples of 9: 9, 18, 27, 36, 45, 54, etc. Notice that all of these sum to 9. Sufficient with C.

I think that by using a little elbow grease, first with writing out your prime possibilities for y, then by writing out the multiples of 9, you can come up with the answer in two minutes and won't be stuck trying to solve this problem. Never hesitate to write out possibilities and see where that gets you. It is very often the right thing to do.