The number of even factors of 21600 is

Question

A. 32
B. 42
C. 60
D. 25
E. 52

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: Make a prime factorization of the number: 21,600 =2^5*3^3*5^2 . According to the above the number of factors is (5+1)(3+1)(2+1)=72 . Now, get rid of powers of 2 as they give even factors --> you'll have 3^3*5^2 which has (3+1)(2+1)=12 factors. All the remaining factors will be odd, therefore 21,600 has 72-12=60 even factors. Answer: C. Similar questions to practice: how-many-odd-positive-divisors-does-540-have-106082.html how-many-of-the-factors-of-72-are-divisible-by-100708.html how-many-even-different-factors-does-the-integer-p-have-132835.html P.S. Please do not reword or shorten questions when posting. Thank you.

Klenex · Answer

Thank you for the answer. But I am not able to figure out the 60 even factors

I have approached like this

As Even*Even=Even & Even*Odd=Even , I considered 2^5*3^3 which has (5+1)(3+1)=24 factors and 2^5*5^2 which has (5+1)(2+1)=18 factors.

So, total 24+18=42 factors. What am I missing here?

Bunuel · Answer

This approach is not correct. Out of 24 factors of 2^5*3^3 not all are even, for example, 3 is a factor of it and is odd, The same for 2^5*5^2. Correct approach is given in my post above.

Shiv2016 · Answer

Total number of factors- Total number of odd factors= Total number of even factors

72-12= 60

arhitsharma · Answer

Is it possible to do the question this way :

1. 21,600 =2^5*3^3*5^2
2. Take one 2 out. so the powers left are 2^{4}*3^{3}*5{2}
3. Number of factors now are : (4+1)*(3+1)*(2+1)
4. 5*4*3 = 60

Bunuel · Answer

No. You got correct answer just by luck. The proper way is given here: https://gmatclub.com/forum/the-number-o ... l#p1311013

ScottTargetTestPrep · Answer

Solution:
 
First, we need to prime factorize 21,600:

21,600 = 216 x 100 = 6^3 x 2^2 x 5^2 = 2^3 x 3^3 x 2^2 x 5^2 = 2^5 x 3^3 x 5^2

The total number of factors of 21,600 is found by adding 1 to each exponent and then calculating the product of those sums. Thus, we see that 21600 has (5 + 1)(3 + 1)(2 + 1) = 6 x 4 x 3 = 72 factors.

To determine the total number of odd factors, we consider only the odd factors of 21,600, which are 3^3 and 5^2. Thus, these two factors generate (3 + 1)(2 + 1) = 4 x 3 = 12 factors that are odd. Therefore, there are 72 - 12 = 60 even factors of 21,600

Answer: C

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The number of even factors of 21600 is

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