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Re: Factors of 36^2 [#permalink]
22 Jan 2012, 16:07

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enigma123 wrote:

How many factors does 36^2 have?

2 8 24 25 26

Answer is 25 i.e. D. Can someone please explain how?

Finding the Number of Factors of an Integer

First make prime factorization of an integer n=a^p*b^q*c^r, where a, b, and c are prime factors of n and p, q, and r are their powers.

The number of factors of n will be expressed by the formula (p+1)(q+1)(r+1). NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: 450=2^1*3^2*5^2

Total number of factors of 450 including 1 and 450 itself is (1+1)*(2+1)*(2+1)=2*3*3=18 factors. For more on number properties check: math-number-theory-88376.html

BACK TO THE ORIGINAL QUESTION: How many factors does 36^2 have? A. 2 B. 8 C. 24 D. 25 E. 26

36^2=(2^2*3^2)^2=2^4*3^4 and according to above it'll have (4+1)(4+1)=25 different positve factros, including 1 and 36^2 itself.

Answer: D.

5 seconds approach: 36^2 is a perfect square. # of factors of perfect square is always odd (as perfect square has even powers of its primes then when adding 1 to each and multiplying them as in above formula you'll get the multiplication of odd numbers which is odd). Only answer choice D is odd thus it must be correct.

Answer: D.

Tips about the perfect square: 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 powers of its prime factors.

Re: Factors of 36^2 [#permalink]
06 Dec 2012, 12:51

Bunuel wrote:

enigma123 wrote:

How many factors does 36^2 have?

The number of factors of n will be expressed by the formula (p+1)(q+1)(r+1). NOTE: this will include 1 and n itself.

Hi Bunuel, can you please clarify when we count multiplies of 1? When do we need to use (1+1)(4+1)(4+1) applicable to this question. Thanks _________________

Re: Factors of 36^2 [#permalink]
07 Dec 2012, 06:15

Bunuel wrote:

Don't understand your question. Can you please elaborate?

I mean 36^2=(2^4)*(3^4), the number of factors is (4+1)(4+1)=25 - this is clear. However 36^2 could be (1^1)(2^4)(3^4), which comes to (1+1)(4+1)(4+1)=50. In which cases in GMAT should I count the powers of 1, which doubles the number of factors? Thank you _________________

Re: Factors of 36^2 [#permalink]
07 Dec 2012, 06:24

Expert's post

fukirua wrote:

Bunuel wrote:

Don't understand your question. Can you please elaborate?

I mean 36^2=(2^4)*(3^4), the number of factors is (4+1)(4+1)=25 - this is clear. However 36^2 could be (1^1)(2^4)(3^4), which comes to (1+1)(4+1)(4+1)=50. In which cases in GMAT should I count the powers of 1, which doubles the number of factors? Thank you

Understood now. The answer is never.

To find the number of factors of a positive integers we should make its prime factorization. Now, 1 is not a prime, thus you shouldn't write it when making prime factorization of an integer.

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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