M12-19

Question

How many positive factors of 72 are divisible by 2?

A. 4
B. 5
C. 6
D. 8
E. 9

Mo2men · Accepted Answer

Hi Ben,

You can do the same concept every time. Let's take your example what are all factors of 252 that are divisible by 7?

Divide 252 by 7 = 36 = 2^2 * 3^2

Total factors in 36 are (2+1) (2+1)= 9

252 = 2^2 * 3^2 * 7. Total factors in 252 are 18.

1 , 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252

We 9 factors that is divisible by 7:   7, 14, 21, 28, 42, 63, 84, 126, 252

Hope it helps

Bunuel · Answer

Official Solution:

How many positive factors of 72 are divisible by 2?

A. 4
B. 5
C. 6
D. 8
E. 9

Finding the Number of Factors of an Integer 
 
First, make the 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 respective 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 
 
The 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:
 
Using the method above, since  72=2^3*3^2 , the total number of factors of 72 is  (3+1)(2+1) = 12 . Of these, only three are odd: 1, 3, and 9. This is because the only factors of 72 that are odd arise from powers of 3. The number of odd factors for  3^2  is  (2 + 1) = 3 , specifically - 1, 3, and 9. Thus, the other  12-3=9  factors are even.
 
OR: Since  72=2^3*3^2 , even factors MUST have 2 raised to either the power of 1, 2, or 3, providing 3 options, and 3 raised to the power of 0, 1, or 2, again giving 3 options. This results in a total of  3*3=9  even factors.

Answer: E

bsattin1 · Answer

Bunuel

Here's how I did it. I'm not sure if this can be generalized, but I think it can be:

If we want all factors of 72 that are divisible by 2, then the factor itself could be thought of as 2 * x = factor of 72. Therefore, x must be a factor of 36. If you wanted a list of factors, all you'd have to do is find the factors of 36 and multiply by 2

with that I get 2^2 * 3^2 = (2 + 1)(2 + 1) = 9.

while it was easy to find factors that are odd, what if the question were what are all factors of 72 that are divisible by 3, or what are all factors of 252 that are divisible by 7?

Cheers,
Ben

nealz · Answer

72=2^3 * 3^2 this gives a tota of 12 factor.

here we have 2^0 =1

all the multiple of 2^0 will be odd.

there will be three multiple of 2^0 with the other prime number in this question which are:
3^0, 3^1, 3^2

Hence 3 numbers will have odd factor.

hence 12-3 = 9 even factor.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

anushree01 · Answer

Hi  Bunuel , I do know how to find the total no. of factors of any given number, but Is there a formula to find its number of even & Odd factors ?
Thanks in advance.

Bunuel · Answer

To find the number of odd factors, simply ignore the powers of 2 in the prime factorization. To find the number of even factors, subtract the odd factors from the total number of factors.

For example, consider 2^3 * 5 * 7^2.

Ignoring 2^3, the remaining part is 5 * 7^2, which has (1 + 1)(2 + 1) = 6 factors. So there are 6 odd factors. 
 The total number of factors of 2^3 * 5 * 7^2 is (3 + 1)(1 + 1)(2 + 1) = 24. 
 Therefore, the number of even factors is 24 - 6 = 18.

Alternatively, to find the number of even factors, don’t add 1 to the power of 2. So for 2^3 * 5 * 7^2 we get 3(1 + 1)(2 + 1) = 18.

Hope it helps.

bot00078 · Answer

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