Tsuruga wrote:
Perfect square has the following properties:
1) The number of distinct factors of a perfect square is ALWAYS ODD.
2) The sum of distinct factors of a perfect square is ALWAYS ODD.
3) A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors.
4) Perfect square always has even number of powers of prime factors.
I think for 4), if a number has even number of powers of prime factors, then it's a perfect square (which means the vice versa statement is correct also). How about 1), 2), 3)? Can anyone confirm?
Tips about perfect squares:
1. The
number of distinct factors of a (positive) perfect square is ALWAYS ODD.
The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square;
2. The
sum of distinct factors of a perfect square is ALWAYS ODD.
The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50;
3. A perfect square ALWAYS has an
ODD number of Odd-factors, and
EVEN number of Even-factors.
The reverse is also true: if a number has an ODD number of Odd-factors, and EVEN number of Even-factors then it's a perfect square. For example: odd factors of 36 are 1, 3 and 9 (3 odd factor) and even factors are 2, 4, 6, 12, 18 and 36 (6 even factors);
4. Perfect square always has
even powers of its prime factors.
The reverse is also true: if a number has even powers of its prime factors then it's a perfect square. For example: \(36=2^2*3^2\), powers of prime factors 2 and 3 are even.
Hope it helps.