If the positive integer N is a perfect square, which of the following must be true?
I. The number of distinct factors of N is odd.
II. The sum of the factors of N is odd.
III. The number of distinct prime factors of N is even.
OA is I and II
My solution is let N = n square, so the factors of N are 1, root n, n, n*root n, and nsquare; in total 5 factors
[spoiler]stmnt I is true(5 is an odd number)
stmnt II - if N is odd, the sum of factors is 1 +odd +odd +odd + odd = odd
if N is even, the sum of factors is 1 + even + even +even +even = odd again, so stmnt II is true
Stmnt III - the number of distinct prime factors is odd. How, if N = 4, its prime factor is only 2(just one = odd no. of factors) if N = 9, its distinct prime factors is only 3. If N is 36, its distint prime no.s are 2,3 (even). Hence stmnt 3 isnt always true.[/spoiler]
let me knw of the method is correct.
MGMAT has a slightly complex method..
I. The number of distinct factors of N is odd.
II. The sum of the factors of N is odd.
III. The number of distinct prime factors of N is even.
OA is I and II
My solution is let N = n square, so the factors of N are 1, root n, n, n*root n, and nsquare; in total 5 factors
[spoiler]stmnt I is true(5 is an odd number)
stmnt II - if N is odd, the sum of factors is 1 +odd +odd +odd + odd = odd
if N is even, the sum of factors is 1 + even + even +even +even = odd again, so stmnt II is true
Stmnt III - the number of distinct prime factors is odd. How, if N = 4, its prime factor is only 2(just one = odd no. of factors) if N = 9, its distinct prime factors is only 3. If N is 36, its distint prime no.s are 2,3 (even). Hence stmnt 3 isnt always true.[/spoiler]
let me knw of the method is correct.
MGMAT has a slightly complex method..
