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

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

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.