summer101 wrote:
Is square root of a positive integer x an integer?
(1) The sum of distinct factors of x is odd
(2) x has an odd number of distinct factor
Tips about the perfect square:
1. The
number of distinct factors of a 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.
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