### 6 - 760+: Solutions - The Patterns Behind those Prior 3 PS Questions

While each of these three prompts might look complex, you CAN get to the correct answer by defining the patterns involved.

1) How many positive integers, from 2 to 100, inclusive, are not divisible by odd integers greater than 1?

In this prompt, we're asked to think about the numbers 2 to 100, inclusive. To start, there's NO way that the GMAT would ask us to truly think about each of these numbers individually, so there MUST be a pattern involved.

Now to the specifics: which of these numbers are NOT divisible by an odd integer that is greater than 1???

Let's start at the first number and work our way "up" until the pattern becomes clear:

2 - this is NOT divisible by any odd integers, so this "fits" what we're looking for...

3 - this IS divisible by an odd integer (3), so it's out

4 - this is NOT divisible by any odd integers, so this "fits"

5 - this IS divisible by an odd integer (5), so it's out

6 - this IS divisible by an odd integer (3), so it's out

7 - this IS divisible by an odd integer (7), so it's out

8 - this is NOT divisible by any odd integers, so this "fits"

Now, looking at the numbers that "fit", we have 2, 4 and 8.....that's 2^1, 2^2 and 2^3....that MUST be the pattern involved, so we can use this knowledge against the rest of the question to find the other values that "fit":

2^4 = 16

2^5 = 32

2^6 = 64

2^7 = 128, but that's outside the range that we were given. Thus, there are 6 values that "fit" what we're looking for.

2) There are 20 doors marked with numbers 1 to 20. And there are 20 individuals marked 1 to 20. An operation on a door is defined as changing the status of the door from open to closed or vice versa. All doors are closed to start with. Now one at a time one randomly picked individual goes and operates the doors. The individual however operates only those doors which are a multiple of the number he/she is carrying. For e.g. individual marked with number 5 operates the doors marked with the following numbers: 5, 10, 15 and 20. If every individual in the group gets one turn, then how many doors are open at the end?

It would take a LOT of time to work through all 20 people and all 20 doors, so I'm going to work through the first several so that we can define the pattern involved...

Remember: All the doors start off CLOSED...

Door 1: Only Person 1 touches this door. So it IS OPEN at the end.

Door 2: Person 1 and Person 2 touch this door. So it is closed at the end.

Door 3: Person 1 and Person 3 touch this door. So it is closed at the end.

Door 4: Person 1, 2 and 4 touch this door. So it IS OPEN at the end.

Now, stop and look at the work that we've done so far... Which doors do we know for sure will be open? Door 1 and Door 4. What do those two numbers have in common? They're both PERFECT SQUARES..... Let's see if that pattern continues...

Door 5: Person 1 and 5 touch this door. CLOSED.

Door 6: Person 1, 2, 3 and 6 touch this door. CLOSED.

Door 7: Person 1 and 7. CLOSED.

Door 8: Person 1, 2, 4 and 8. CLOSED

Door 9: Person 1, 3 and 9. OPEN.

Notice how the next door that we know will be open in the end is Door 9. It is ALSO a PERFECT SQUARE. Given the work we've done so far, this MUST be the pattern, so we're ultimately looking for the number of perfect squares from 1 to 20. They are 1, 4, 9 and 16. That's a total of 4 open doors at the end.

3) A test has 200 questions. Each question has 5 options, but only 1 option is correct. If test-takers mark the correct option, then they are awarded 1 point. However, if an answer is incorrectly marked, the test-taker loses 0.25 points. No points are awarded or deducted if a question is not attempted. A certain group of test-takers attempted different numbers of questions, but each test-taker still received the same net score of 40. What is the maximum possible number of such test-takers?

From the answer choices, we can see that there are a lot of different ways to get a total of 160 points (at least 31 ways...), so there's no way that the GMAT would require that we calculate each individual option. There has to be a pattern, so let's start off with the easiest 'ways' to get 160 points and go from there... Remember - a correct answers get you 1 point, an incorrect answer gets you MINUS 1/4 point and a skipped question gets you 0 points. So, how can we get 160 points...

40 correct, 0 incorrect, 160 skipped

41 correct, 4 incorrect, 155 skipped

42 correct, 8 incorrect, 150 skipped

Etc.

To find all of the options, you can either "count up" (40+0 = 40, 41+4 =45, 42+8 = 50, etc.) OR you can "count down" from the number of skipped questions. (160, 155, 150, etc.). With either option, you'll eventually run out of questions, then you'll be done.

Counting down from 160 is probably faster, as long as you don't forget the final option - the one with 0 skipped questions). Here, there are 33 multiples of 5 (including the 0

'option').

As you continue to study, it's important to remember that most GMAT Quant questions are NOT 'testing' your knowledge of advanced math, specialty formulas or complex ideas. The 'math' behind most Quant questions is actually rather straight-forward. As such, to maximize your score, you have to be ready to 'play around' with 'tough-looking' prompts and find the simple math behind them.

GMAT assassins aren’t born, they’re made,

Rich

If you have any questions about anything in this thread, then you can feel free to contact me directly via email (at [email protected])