GMAT PREP PROBLEM: if x is the product of the integers...

[lindaweichenchu]

If x is the product of the integers from 1 to 150, inclusive, and 5^y is a factor of x, what is the greatest possible value of y?

Below is my solution:

5^1: 5, 10, 15,20,.....,150 => 30 possibilities
5^2: 25, 50, 75, 100, 125,150 => 6 possibilities
5^3 : 125=> 1 possibility

30+6+1=37 ...................and 37 is indeed the answer.

However, I am not sure why I did not double/triple counted figures such as 25, 50,...125.

Experts, please help!
Thanks a lot!

[jnelson0612]

If x is the product of the integers from 1 to 150, inclusive, and 5^y is a factor of x, what is the greatest possible value of y?

Below is my solution:

5^1: 5, 10, 15,20,.....,150 => 30 possibilities
5^2: 25, 50, 75, 100, 125,150 => 6 possibilities
5^3 : 125=> 1 possibility

30+6+1=37 ...................and 37 is indeed the answer.

However, I am not sure why I did not double/triple counted figures such as 25, 50,...125.

Experts, please help!
Thanks a lot!

Hi Linda,
Sure! Okay, so if 5^y is a factor of x, we just want to know how many factors of 5 x contains.

You are correct that there are 30 multiples of 5 in x, so there are 30 factors of 5 there (your 5^1 list).

Now, some of these factors have an extra 5: 25, 50, 75, 100, 125, 150. Six numbers contain an extra 5. Notice that you already counted the first factor of 5 in your 5^1 count; this is just counting the extra factors of 5. So there are six additional fives when we consider the 5^2 numbers.

And one number, 125, has one additional 5, since it is 5^3. Again, notice, you counted one 5 in your 5^1 list, and another 5 in your 5^2 list. You have already counted those, so we only need to count one more 5 for the 5^3.

Thus, 30 + 6 + 1 = 37.

Congratulations!

[lindaweichenchu]

Thanks for clearing up my blind spots and that totally makes sense. Thanks Jamie!!

[tim]

:)