Lesson: Squares of Integers

Comment on Squares of Integers


I have a question regarding counting. I think I'm confusing some concepts here. As the value is less than 81 then we know that the values are indeed from 0 to 80 inclusive. In that case should we be calculating the integers as (80-0)+1?
greenlight-admin's picture

Question link: https://gre.myprepclub.com/forum/set-s-consists-of-all-positive-integers...

Yes, there are 81 digits from 0 to 80 inclusive.
This means there are 80 digits from 1 to 80 inclusive.

However, this doesn't mean there are 80 numbers in set S, because "set S consists of all positive integers less than 81 that are NOT EQUAL TO THE SQUARE OF AN INTEGER"

So, we must remove all perfect squares from our collection of integers from 1 to 80 inclusive.

Does that help?


Dear Brent
I am a little confused.
80 = 2x2x2x2x5
= 2^4x 5^1
= (4+1)(1+1)
= (5)(2)= 10

81 = 3^4
= (4+1) = 5

How we got 8 to deduct from 80 to get 72?
your reply is appreciated.
greenlight-admin's picture

Question link: https://gre.myprepclub.com/forum/set-s-consists-of-all-positive-integers...

I think you might be misreading the question.
Your calculations are for finding the number of positive divisors of 80 and 81, but the question doesn't ask us about the number of positive divisors.

GIVEN: Set S consists of all positive integers less than 81 that are not equal to the square of an integer.

Here are some SQUARES of integers: 1, 4, 9, 16, 25, etc..
Notice that:
1 = 1²
4 = 2²
9 = 3²
16 = 4²

So, Set S consists of all positive integers less than 81 EXCEPT for 1, 4, 9, 16, etc.
In other words, Set S = {2,3,5,6,7,8,10,11,12,13,14,15,17,18.......78,79,80}

Does that help?

Yes thanks. I just missed the line "positive integers" and therefore counting from 0 instead of 1.

At 2:50 I was trying to figure out why there was an odd number of divisors, and I noticed each divisor had a corresponding number which made a pair (like 1 and 16, 2 and 8), except for the middle number. Because, of course, since we are talking about perfect squares one of it's divisors will be a number times itself, like 4x4, 7x7, 6x6. Since repeats aren't counted when counting divisors, then the number of divisors will be odd for a perfect square; that is, because one pair, will actually only consist of one unique number. Anyway, for me, I thought it was another good way to remember this.
greenlight-admin's picture

That's a great way to remember that squares of integers have an odd number of positive divisors.
Nice work!

Hi Brent,

Can you provide the solution to this question?

greenlight-admin's picture

Thanks for the heads up!
I didn't realize I had forgotten to answer that.
Here's my full solution: https://gre.myprepclub.com/forum/positive-integer-n-has-k-positive-divis...


Hi Brent,


For above question why the answer is 12? Can't n=144 for example?
Question doesn't mention anything about upper bound for the integer.
greenlight-admin's picture

Question link: https://gre.myprepclub.com/forum/if-n-is-a-positive-integer-and-n-2-is-d...

You're correct to say that n COULD equal 144.
However, the question asks us to find the largest positive integer "that MUST divide into n"
In other words, we're looking for the biggest number that MUST be a factor of n.

Notice that n = 12 satisfies the condition that n² is divisible by 72.
Since 144 is not a factor of 12, we can't say that 144 must be a factor of n.

Does that help?

I am glad I found this courses to guide me through my success to GRE. Thank you very much.
However, I really struggle to understand the aforementioned question...Could you enlighten me further please???
greenlight-admin's picture

My solution: https://gre.myprepclub.com/forum/if-n-is-a-positive-integer-and-n-2-is-d...

Hi Kristina,
I'm glad you like the course!

I'm having a hard time coming up with a solution that's different from my original solution.
Is there a specific point in my solution that you're having trouble understanding?

In the meantime, you might want to review some of the solutions to the same question on GMAT Club: https://gmatclub.com/forum/if-n-is-a-positive-integer-and-n-2-is-divisib...
Perhaps you'll find an approach that better resonates with you.

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