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# How many factors does 36^2 have?

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How many factors does 36^2 have? [#permalink]

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Updated on: 31 Jul 2012, 02:53
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How many factors does 36^2 have?

A. 2
B. 8
C. 24
D. 25
E. 26

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Originally posted by enigma123 on 22 Jan 2012, 16:48.
Last edited by Bunuel on 31 Jul 2012, 02:53, edited 1 time in total.
Edited the question and added the OA.
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22 Jan 2012, 17:07
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enigma123 wrote:
How many factors does 36^2 have?

2 8 24 25 26

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.

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.

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.

Hope it helps.
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Re: How many factors does 36^2 have? [#permalink]

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29 Nov 2012, 03:05
valerjo79 wrote:
How many factors does 36^2 have?

Does anyone see a fast solution?

Thanks

2
8
24
25
26

$$36^2 = (2^2 * 3^2)^2 = 2^4*3^4$$

Hence number of factors = ($$4+1) * (4+1) = 25$$.

Ans D it is!
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06 Dec 2012, 13:51
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
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07 Dec 2012, 02:15
fukirua wrote:
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

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07 Dec 2012, 07:15
Bunuel wrote:

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
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07 Dec 2012, 07:24
fukirua wrote:
Bunuel wrote:

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.

Hope it's clear.
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Re: How many factors does 36^2 have? [#permalink]

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01 May 2014, 21:18
I remember reading a rule in MGMAT that any perfect square will have an odd number of factors. 25 is the only odd answer choice. Is that right?
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Re: How many factors does 36^2 have? [#permalink]

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02 May 2014, 01:51
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lmariesf wrote:
I remember reading a rule in MGMAT that any perfect square will have an odd number of factors. 25 is the only odd answer choice. Is that right?

Yes, it is. Check my solution here: how-many-factors-does-36-2-have-126422.html#p1032696

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.

Hope this helps.
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Re: How many factors does 36^2 have? [#permalink]

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25 Sep 2017, 17:59
enigma123 wrote:
How many factors does 36^2 have?

A. 2
B. 8
C. 24
D. 25
E. 26

We can use the rule of adding 1 to each exponent of each unique prime and then multiplying our values together:

36^2 = (9 x 4)^2 = (3^2 x 2^2)^2 = 3^4 x 2^4

(4 + 1)(4 + 1) = 5 x 5 = 25

Alternate solution:

An interesting fact about perfect squares greater than 1 is that they always have an odd number of factors. For example, the factors of 4 are 1, 2, and 4 (a total of 3 factors) and the factors of 100 are 1, 2, 5, 10, 20, 50, and 100 (a total of 7 factors). Since 36^2 is a perfect square, it should have an odd number of factors. The only odd number in the answer choices is 25.

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Re: How many factors does 36^2 have? [#permalink]

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29 Jan 2018, 13:13
Isn't it supposed to be how many positive factors does 36^2 have?
Otherwise, the answer should be 50.
(-36)*(-36)= 36^2 as well. ect
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Re: How many factors does 36^2 have? [#permalink]

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29 Jan 2018, 21:19
mahagmat wrote:
Isn't it supposed to be how many positive factors does 36^2 have?
Otherwise, the answer should be 50.
(-36)*(-36)= 36^2 as well. ect

A factor is a POSITIVE divisor.
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Re: How many factors does 36^2 have? [#permalink]

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30 Jan 2018, 03:58
Why is that ?
For example, 36= 36 x 1 ( 36 and 1 are factors of 36)
But, 36= -36 x -1 ( -36 and -1 are also factors of 36)
I came across a question once that said : What is the sum of all factors of X, the answer was 0, which for me was corehent.

On magoosh, the mention both negative and positive factors :
"For the GMAT: The same rules in the GRE apply to the GMAT in this case. Once again, the factors and multiples of an integer include both the positive and negative integers/multiples.
Once again, we haven't seen an official question that involved negative factors or multiples, but it doesn't mean that they don't exist. However, like the GRE, the questions will also clearly specify "positive factors" or "positive integers". :D"

Can you please explain how do you only consider positive factors in this case?
Thank you.
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How many factors does 36^2 have? [#permalink]

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30 Jan 2018, 11:14
Quote:

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.

Hope it helps.

Hello niks18, could you please advice regarding Bunuels tips about the perfect square. please see below and advice what i missed or did wrong, or counted incorrectly many thanks!

lets take 36. here is prime factorization of 36 (2*18) ---> (18) ---> (2*9) ---> (9) --->(3*3)

$$Tips about the perfect square$$

1. The number of distinct factors of a perfect square is ALWAYS ODD. lets take 36 in all cases to test your tips:) - it has 1, 2, 3, 9 = 4 as distinct factors, no but it is EVEN

2. The sum of distinct factors of a perfect square is ALWAYS ODD. 36 - so it has 1, 2, 3, 9 as distinct factors hence 1+ 2+3+9 = 15 is it correct ?

3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. let me break down this sentence into two clauses:)

A perfect square ALWAYS has an ODD number of Odd-factors ---> odd numbers are 1, 3, 9 so total 3 odd numbers

A perfect square ALWAYS has an EVEN number of Even-factors ---> 2, 8 so in total 2 numbers is corrrect but how about 6 ! ?? or is my prime factorization incorrect ??

4. Perfect square always has [b]even powers of its prime factors
. ---> so $$36$$ = $$2^2$$ and $$3^2$$
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Re: How many factors does 36^2 have? [#permalink]

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30 Jan 2018, 11:36
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dave13 wrote:
Quote:

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.

Hope it helps.

Hello niks18, could you please advice regarding Bunuels tips about the perfect square. please see below and advice what i missed or did wrong, or counted incorrectly :) many thanks!

lets take 36. here is prime factorization of 36 (2*18) ---> (18) ---> (2*9) ---> (9) --->(3*3)

$$Tips about the perfect square$$

1. The number of distinct factors of a perfect square is ALWAYS ODD. lets take 36 in all cases to test your tips:) - it has 1, 2, 3, 9 = 4 as distinct factors, no but it is EVEN :?

2. The sum of distinct factors of a perfect square is ALWAYS ODD. 36 - so it has 1, 2, 3, 9 as distinct factors hence 1+ 2+3+9 = 15 is it correct ?

3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. let me break down this sentence into two clauses:)

A perfect square ALWAYS has an ODD number of Odd-factors ---> odd numbers are 1, 3, 9 so total 3 odd numbers

A perfect square ALWAYS has an EVEN number of Even-factors ---> 2, 8 so in total 2 numbers is corrrect :? but how about 6 ! ?? or is my prime factorization incorrect ??

4. Perfect square always has [b]even powers of its prime factors
. ---> so $$36$$ = $$2^2$$ and $$3^2$$

Hi dave13

you have factorized 36 incorrectly.

$$36=2^2*3^2$$. Hence number of factors $$= (2+1)*(2+1)=9=Odd$$

Factors of 36 are 1,2,3,4,6,9,12,18 & 36. Now figure out the validity of tips yourself.
Re: How many factors does 36^2 have?   [#permalink] 30 Jan 2018, 11:36
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