How Many Even Divisors of 1600 are Not Multiples of 16?

Question - How many even divisors of 1600 are not multiples of 16?

(A) 4
(B) 6
(C) 9
(D) 12
(E) 18

“How many even divisors of 1600 are not multiples of 16?” – is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "The Official Guide for GMAT Review". GMAT Quant section consists of a total of 31 questions. GMAT Problem Solving questions consist of a problem statement followed by five options. GMAT Problem solving comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Solution and Explanation:

Approach 1:

We need to prime factorize 1600:
1600 = 16*100
=>2^6*5*2

Total no of divisors of 1600 = (6+1)*(2+1) = 7*3 = 21
Out of which 1, 5 & 25 are odd divisors
Total no of even divisors 0f 1600 = 21-3 = 18
1600 = 2^4 (2^2*5^2)
1600 = 16 (2^2*5^2)
No of divisors which are multiple of 16 = (2+1)*(2+1) = 3*3=9 all are even
No of even divisors which are not multiple of 16 = 18 - 9 = 9

Similarly,
2^0, 2^4, 2^5, 2^6 are not allowed
Only 2^1, 2^2, 2^3 are allowed.
Hence, we get 3 ways.

Also,
5^0, 5^1 and 5^2 are allowed, which are three ways.
Total no of even divisors not multiple of 16 = 3*3 =9

Hence, C is the correct answer.

Approach 2:

For this manual method we again need to prime factorize 1600:
1600=16∗100=
=>2^6*5*2

So even factors not devisibe by 16 would be the product of 2's and 5's we have when the highest possible power of 2 is 3.
Let's manually find those factors:

1. 2
2. 2^2
3. 2^2
4. 2*5
5. 2^2*5
6. 2^3*5
7. 2*5^2
8. 2^2*5^2
9. 2^3*5^2

Hence, 9 is the correct answer.

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