BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

Kaplan: Properties of a Square

This topic has expert replies
Post new topic Post Reply
Previous Topic Next Topic
Senior | Next Rank: 100 Posts
Posts: 72
Joined: Mon Apr 11, 2011 6:45 pm
Thanked: 2 times

Kaplan: Properties of a Square

by shubhamkumar » Sun Apr 15, 2012 9:29 am
Which of the following must be true if the square root of X is a positive integer?

I. X has an even number of distinct factors.

II. X has an odd number of distinct factors.

III. The sum of X's distinct factors is odd.

A.I only
B.II only
C.I and III
D.II and III
E.I, II, and III

OA: D
Top
Source: — Problem Solving |
Senior | Next Rank: 100 Posts
Posts: 92
Joined: Thu Oct 06, 2011 8:06 am
Thanked: 18 times

by Neo Anderson » Sun Apr 15, 2012 9:46 am
It's the property of a perfect square that: it has only odd number of distinct factors and sum of all these factors is odd.
take example of 4; 3 factors: 1,2,4; sum = 1+2+4 = 7 odd
9; 3 factors: 1,3,9; sum = 13 odd
16; 5 factors: 1,2,4,8,16; sum = 31 odd
25; 3 factors: 1,5,25; sum = 31 odd again
Top
User avatar
Legendary Member
Posts: 588
Joined: Sun Oct 16, 2011 9:42 am
Location: New Delhi, India
Thanked: 130 times
Followed by:9 members
GMAT Score:720

by rijul007 » Sun Apr 15, 2012 9:53 am
shubhamkumar wrote:Which of the following must be true if the square root of X is a positive integer?

I. X has an even number of distinct factors.

II. X has an odd number of distinct factors.

III. The sum of X's distinct factors is odd.

A.I only
B.II only
C.I and III
D.II and III
E.I, II, and III

OA: D
Lets say X = a^2 * b^2 * c^2 * d^2
where, a,b,c and d are prime factors
Then number of distict factors = (2+1)*(2+1)*(2+1)*(2+1) = 81

II is true
I is not true

Eliminate A, C and E

lets take the same value, X = a^2 * b^2 * c^2 * d^2
lets say a=2
no of odd factors = (2+1)*(2+1)*(2+1)=27
so of 27 odd factors = odd
sum of all factors = odd + (sum of 54 even factors) = odd

III is true


II and III must be true


Here are some examples,
Factors of 4 = 1,2,4 (Sum = 7)
Factors of 9 = 1,3,9 (Sum = 13)
Factors of 16 = 1,2,4,8,16 (Sum = 31)


Option D
Top
User avatar
Master | Next Rank: 500 Posts
Posts: 134
Joined: Fri Apr 06, 2012 3:11 am
Thanked: 35 times
Followed by:5 members

by Shalabh's Quants » Sun Apr 15, 2012 10:13 am
shubhamkumar wrote:Which of the following must be true if the square root of X is a positive integer?

I. X has an even number of distinct factors.

II. X has an odd number of distinct factors.

III. The sum of X's distinct factors is odd.

A.I only
B.II only
C.I and III
D.II and III
E.I, II, and III

OA: D
Lets say X = a^x * b^y * c^z
where, a,b,c are prime factors

Then number of distict factors = (x+1)*(y+1)*(z+1)

Take few examples of perfect square nos. 4, 25, 196...

4 = {1,2,4}; No. of factors--odd.; Sum of all factors= 7 Odd

25 = {1,5,25}; No. of factors--odd.;Sum of all factors= 31 Odd

196 = 2^2*7^2; No. of factors = (2+1)*(2+1)= 9 (odd). Statement II is correct.

Sum of all factors = (a^x+1 - 1)/(a-1)*(b^y+1 - 1)/(b-1)*(c^z+1 - 1)/(c-1)

For 196; Sum of all factors= (2^2+1 - 1)/(2-1)*(7^2+1 - 1)/(7-1)= (2^3 - 1)*(7^3 - 1)/6 = 399 (Odd);

Statement II & III both are correct.
Shalabh Jain,
e-GMAT Instructor
Top
Post new topic Post Reply
• Page 1 of 1