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PS approach

by GMAT Kolaveri » Sun Apr 01, 2012 4:58 am
For all positive integers f, fâ—Ž equals the distinct pairs of positive integer factors. For example, 16â—Ž = 3, since there are three positive integer factor pairs in 16: 1 x 16, 2 x 8, and 4 x 4.

What is the greatest possible value for fâ—Ž if f is less than 100?

My question is how to approach such question. From where does one start?

[spoiler]AO:6[/spoiler]
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by Pharo » Sun Apr 01, 2012 5:57 am
I think one way to solve these type of questions is using logic. How can you increase the integer factor pairs? By choosing a number that would be divisible by 2,3,5,7,11 etc; (i.e. the prime numbers). 7 and 11 do not make much sense to me. So I focused on 2,3 and 5. What is a big number that can be divisible by the above three and their multiples? 90 or 60. I started with 60.

1x60
2x30
3x20
4x15
5x12
6x10

6 pairs. Now i thought 90 is bigger! Let's do that too:

1x90
2x45
3x30
5x18
6x15
9x10

6 pairs. Just to make sure; I tried 70:

1x70
2x35
5x14
7x10

4 pairs. So I said the answer is 6 :)

Also, I knew that the Sumerians used a base 60 numerical system since 60 is divisible by many numbers (in those ages where the people did not know much math; this helped them a lot!). That is why i started with 60 instead of 90 :P
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