How man y positive integers less than 50 have an odd number of

Question

How man y positive integers less than 50 have an odd number of positive integer divisors?

(A) 3
(B) 5
(C) 7
(D) 9
(E) 11

Are You Up For the Challenge: 700 Level Questions

IanStewart · Accepted Answer

For most numbers, their divisors come in pairs. If you look at 10, say, we can write its four divisors in pairs that give us 10 as a product:

1 * 10 = 10
2 * 5 = 10

The only kind of number that has one unpaired divisor is a square -- if you look instead at 36, say, we can write most of its divisors in pairs just as above:

1*36 = 36
2*18 = 36
3*12 = 36
4*9 = 36

but then we have one divisor left, the square root of 36:

6^2 = 36.

so 36 has an odd number of divisors, because it's equal to the square of a smaller integer. So it's only perfect squares that have an odd number of divisors in total, and the question here is just asking how many nonzero squares there are less than 50, so the answer is seven (the squares from 1^2 through 7^2 are less than 50).

zhanbo · Answer

Notice that the number of divisors of a number is the product of each exponent plus one.
They must be squares, so the answer is ⌊sqrt(50)⌋=7

Answer is  (C) .

Devoshish · Answer

didnt understand this can you please explain how you got the solution.

PVOG · Answer

This is one of those cases where knowing the properties of square numbers will help you solve quickly. I am not great at quant, my score only sits around Q49, yet I myself was able to solve this in 26 seconds with some basic knowledge of number properties, so let me try and share my thinking to help you.

All square numbers have an odd number of factors (because one of the factor pairs is actually an integer repeated twice).

E.g. factor pairs of 36 are (1,36), (2,18), (3,12), (4,9),(6,6)

The (6,6) is repeated, the number of distinct factors is ODD for all square numbers as the above example shows. Now, what this question is asking is how many positive integers less than 50 have an odd number of positive divisors, in other words, how many positive integers less than 50 have an odd number of total factors.

Translation: how many square numbers are there between 0 and 50...
 
Well, in order to have an odd number of total factors, the number must be a square number, so the numbers 1,4,9,16,25,36, and 49 all qualify, that is, a total of 7 numbers.

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

How man y positive integers less than 50 have an odd number of

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

How man y positive integers less than 50 have an odd number of

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards