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How many factors of 4500 have odd number of factors?

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GMATinsight
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How many factors of 4500 have odd number of factors? [#permalink] New post   19 Jan 2020, 06:51
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00:00
A
B
C
D
E

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85% (hard)

Question Stats:

34% (01:42) correct 66% (01:35) wrong based on 68 sessions
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How many factors of 4500 have odd number of factors?

A) 4
B) 6
C) 8
D) 12
E) 16

Source: https://www.GMATinsight.com
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C
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Re: How many factors of 4500 have odd number of factors? [#permalink] New post   19 Jan 2020, 07:28
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4500= (2^2)*(3^2)*(5^3)
For factors having odd factors, factor must have even powers.
So such factors are 1, 2^2, 3^2, 5^2, (2^2)*(3^2), (3^2)*(5^2), (2^2)*(5^2), (2^2)*(3^2)*(5^2) - Total 8 factors
1 is considered because 1 has only one factor that is 1.
C is answer.
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Re: How many factors of 4500 have odd number of factors? [#permalink] New post   20 Jan 2020, 04:17
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GMATinsight wrote:
How many factors of 4500 have odd number of factors?

A) 4
B) 6
C) 8
D) 12
E) 16

Source: https://www.GMATinsight.com


Property: Perfect squares (i.e. squares of Integers) always have odd number of Factors. See ref: https://gmatclub.com/forum/how-many-int ... l#p1549663

i.e. All factors of 4500 which are squares of some integers need to be counted to answer this question

\(4500 = 2^2*3^2^5^3\)

So the factors made up by \(2^0\), \(2^2\), \(3^0\), \(3^2\), \(5^0\), \(5^2\),

We can take one number from each of the three colour codings and make combinations such as

\(2^0*3^0*5^0\)
\(2^0*3^2*5^0\)
\(2^2*3^0*5^0\)
\(2^2*3^2*5^0\)

\(2^0*3^0*5^1\)
\(2^0*3^2*5^1\)
\(2^2*3^0*5^1\)
\(2^2*3^2*5^1\)

i.e. Total combinations = 2*2*2 = 8 (as mentioned above

Answer: Option C
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Re: How many factors of 4500 have odd number of factors? [#permalink] New post   21 Jan 2020, 22:20
sorry, I still cannot understand both described methods to answer the question. Can someone advise theory in which this example can be explained in more details ?

thank you,
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Re: How many factors of 4500 have odd number of factors? [#permalink] New post   29 Nov 2022, 11:58
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Re: How many factors of 4500 have odd number of factors? [#permalink]
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