4k + 2 = 2(2k + 1) = 2*odd. So, we are looking for even factors which are not multiples of 4.

180 = 2^2*3^2*5. Consider the part without 2^2. Now, 3^2*5 has (2 + 1)(1 + 1) = 6 factors: 1, 3, 5, 9, 15, 45. Any of them paired with 2 will be even factor of 180 which is not a multiple of 4, so 2, 6, 10, 18, 30 and 90.

Answer: E.
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Bunuel how did you find 6 factors as 1,3,5,9,15,45. please help me in this.

I just manually listed the factors of 90 = 3^2*5. You don't have to find the factors for this question though, it's done here just to illustrate. The fact that 90 has 6 factors (does not matter what they really are) is enough to get that 180 will have twice as many even factors which are not multiples of 4.
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How did you got (2+1)(1+1) ?? Pls explain

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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.

2. Properties of Integers

• Theory
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5. Divisibility/Multiples/Factors

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  3 Deadly Mistakes you must avoid in LCM-GCD Questions
  How Well Do You Know Your Factors?
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    The E-GMAT Primes Trio - 3 Ques on No. of factors & Prime factors
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For other subjects:
ALL YOU NEED FOR QUANT ! ! !
Ultimate GMAT Quantitative Megathread
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There are 6 possibilities, where f(x)=4x+2 is a factor of 180.

f(0)=2
F(1)=6
f(2)=10
f(4)=18
f(7)=30
f(22)=80

I did this question with simple substitution like below -

we know K >=0 and is an integer.

therefore for
K = 0 ; (4*k + 2) = 2 - A factor of 180
K = 1 ; (4*k + 2) = 6 - A factor of 180
K = 2 ; (4*k + 2) = 10 - A factor of 180
K = 4 ; (4*k + 2) = 18 - A factor of 180

No we know answer can be D or E.

If I am able to find one more my answer will be E.

Hence I factorized 180 = 2x3x3x2x5

Next number after 18 I can form is 30 that will be a factor of 180 and I get that with k = 7.

Hence E.

FYI - I did this question under two minutes. So IMO not a lengthy method.
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