Problem Solving

Hi All,

We're told that Mr. McCall selects a number that is two-digit and positive: If the number is prime, then he assigns that many problems for homework. If the number is NOT prime, then he assigns 8 MORE problems than the number for homework. We're told he assigns 97 problems for homework. We're asked which of the following could be the original number he selected. This question can be solved in a couple of different ways, including by TESTing THE ANSWERS.

To start, it's worth noting that there are only TWO potential ways to get to a specific number of questions, so only 1 or 2 of the three given values will 'fit' what we're told.

I. 89
The number 89 is prime. You can prove it by doing a little division. Since 89 is ODD, no even numbers will divide evenly into it and since it ends in '9', we know that 5 does not divide evenly in. Thus, you really just have to check 3, 7 and 9. The sum of the digits 8+9 = 17, so using the 'rule of 3' and the 'rule of 9', you can quickly eliminate those options. That just leaves the 7 - and 7 does NOT divide evenly into 89. Thus, Mr. McCall would have assigned 89 questions if he had chosen 89; he assigned 97 questions though, so this number is NOT correct.
Eliminate Answers A, B and E.

II. 97
Since Roman Numeral II is in both of the remaining answers, we KNOW that it's a valid option. To be thorough though, the number 97 is prime - and you can prove it in the exact same way that we proved that 89 is prime (see above).

III. 105
The number 105 is NOT prime; it actually has a number of different factors (the most obvious of which is 5, since 105 ends in a '5'). Thus, Mr. McCall would have assigned 8 MORE - meaning 105+8 = 113 questions - in this situation. This does NOT match the 97 questions that we're told he assigns though, so this number is NOT correct.
Eliminate Answer D.

Final Answer: C

GMAT assassins aren't born, they're made,
Rich