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Vice versa rule for Perfect Square

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Intern
Joined: 11 Feb 2015
Posts: 5
Vice versa rule for Perfect Square  [#permalink]

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04 Jun 2015, 02:21
1
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?
Math Expert
Joined: 02 Sep 2009
Posts: 58313
Re: Vice versa rule for Perfect Square  [#permalink]

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04 Jun 2015, 02:37
1
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?

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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Joined: 11 Feb 2015
Posts: 5
Re: Vice versa rule for Perfect Square  [#permalink]

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04 Jun 2015, 02:43
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?

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!
CEO
Joined: 20 Mar 2014
Posts: 2603
Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
GMAT 1: 750 Q49 V44
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WE: Engineering (Aerospace and Defense)
Re: Vice versa rule for Perfect Square  [#permalink]

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04 Jun 2015, 03:20
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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29 Jan 2018, 00:20
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Re: Vice versa rule for Perfect Square   [#permalink] 29 Jan 2018, 00:20
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