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Vice versa rule for Perfect Square

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Vice versa rule for Perfect Square [#permalink]

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New post 04 Jun 2015, 01:21
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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? :)
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Re: Vice versa rule for Perfect Square [#permalink]

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New post 04 Jun 2015, 01:37
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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.
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Re: Vice versa rule for Perfect Square [#permalink]

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New post 04 Jun 2015, 01:43
Bunuel wrote:
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.


Awesome and right to the point! Thanks Bunuel!
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Re: Vice versa rule for Perfect Square [#permalink]

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New post 04 Jun 2015, 02:20
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? :)



Try and run some scenarios instead on learning abstract rules. Try 9,16,25,36 for properties of perfect squares and pick 8 or 12 or 20 for non perfect squares just to compare the properties and see the pattern for your questions above. I feel this way you won't have to learn so many of these formulae.
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Re: Vice versa rule for Perfect Square [#permalink]

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New post 28 Jan 2018, 23:20
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Re: Vice versa rule for Perfect Square   [#permalink] 28 Jan 2018, 23:20
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