Number of Even Factors of 84? GMAT Problem Solving

Question: Number of even factors of 84?

1. 2
2. 4
3. 8
4. 10
5. 12

Correct Answer: C
Solution and Explanation:
Approach Solution 1:

84=2² × 3×7
The number of factors:(2+1)*(1+1)*(1+1)=12
3,7,21,and 1 are odd.
So,12-4=8

Hence the correct answer is C.

Approach Solution 2:

If N= P^a*Q^b*R^c and let's assume Q and R are odd prime numbers
Then we know that total number of factors are : (a+1)*(b+1)*(c+1) [A]
Number of odd factors are: (b+1)* (c+1)
Number of even factors can be found in 2 ways

Option 1:

Subtracting odd number factors from total number factors. Which is [A-B]

Option 2:

a*(b+1)*(c+1) [here a is the power of even factor P]
Now coming to the problem:
84= 2^2*3*7

Applying option 1 approach:

Total number of factors:(2+1)*(1+1)*(1+1)=12
Number of odd factors: 4
Therefore, number of even factors: 12-4=8

Applying option 2 approach:
Number of even factors: 2*(1+1)*(1+1)=8

Hence the correct answer is C.

Approach Solution 3:

Case 1:

84 = 2 * 2 * 3 * 7
Number of even factors: 2, 4, 6, 14, 24, 28, 42, 84 = 8

Hence the correct answer is C.

Case 2:

That implies 84 = 22∗31∗71
= Total factors: (2 + 1) * (1 + 1) * (1 + 1) = 3 * 2 * 2 = 12

Odd factors: 1, 3, 7, 21 = 4
Even factors: 12 - 4 = 8

Hence the correct answer is C.

“Number of even factors of 84?” - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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