How many factors does 36^2 have?

Question

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

Bunuel · Accepted Answer

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 . Hope it helps.

Bunuel · Answer

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.

Back to the original question:

How many factors does 36^2 have?

36^2=(2^2*3^2)^2=2^4*3^4  --> # of factors  (4+1)*(4+1)=25 .

Answer: D.

Or another way: 36^2 is a perfect square, # of factors of perfect square is always odd (as perfect square has even powers of its primes and when adding 1 to each and multiplying them as in above formula you'll get the multiplication of odd numbers which is odd). Only odd answer in answer choices is 25.

Hope it helps.

praveengmat · Answer

Thanks a ton !!.. loved the approach !

AtifS · Answer

Factors of a perfect square can be derived by using prime factorization and then using the formula to find perfect square's factors.

In this case \((36)^2= (2^2*3^2)^2=2^4*3^4\) or \((36)^2=(6^2)^2=(6)^4=(2*3)^4=2^4*3^4\)

And now you can use the formula explained above by Bunuel to determine the answer, which is \((4+1)*(4+1)=5*5=25=Odd\)(Trick is there must be odd number of factors of a perfect square and only 25 is odd in answer choices, so it can be solved within 30 seconds or less

)

Please! go through the GMAT Math Book by GMAT CLUB (written by bunuel & walker), all of these tips & tricks are written there. (even I have compiled them in one .pdf file and is shared here on Math forum)

) Please! go through the GMAT Math Book by GMAT CLUB (written by bunuel & walker), all of these tips & tricks are written there. (even I have compiled them in one .pdf file and is shared here on Math forum)

kuttingchai · Answer

This method is worth bookmarking. Appreciate it

fukirua · Answer

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

Bunuel · Answer

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

Bunuel · Answer

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.

luckyducky7 · Answer

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?

Bunuel · Answer

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

ScottTargetTestPrep · Answer

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. 
 
Answer: D

dave13 · Answer

Hello niks18 , could you please advice regarding Bunuel`s 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 even powers of its prime factors . ---> so 36 = 2^2 and 3^2

niks18 · Answer

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.

Archit3110 · Answer

factors of 36 ; 2^2*3^2 = 9
so for 36^2 ; (2^2)^2*(3^2)^2 ; 2^4*3^4 ; 25
IMO D; 
also another key take away perfect square have odd factors so out of given option only D is valid

sm1510 · Answer

Question seen in Manhattan GMAT test

Abhishek009 · Answer

36^2 = 6^4 = 2^4*3^4

Thus, no of factors must be  (4+1)(4+1) = 25 , Answer must be (D)

shanafairweather · Answer

At first I thought the result should be 1296. But thanks to your explanation, I understand better why 25 comes out. Mathematics is truly magical, isn't it?

bumpbot · Answer

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How many factors does 36^2 have?

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How many factors does 36^2 have?

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