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# How many integers less than 100 have exactly 4 odd factors b

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Senior Manager
Joined: 13 Oct 2016
Posts: 367
GPA: 3.98
How many integers less than 100 have exactly 4 odd factors b  [#permalink]

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31 Oct 2016, 02:43
1
In order to achieve requested 4 factors the number should be in the form p^3 or p*q (giving 3+1 and 2+2 total number of factors), where p, and q are odd prime factors.
In first case we have only one match - 3^3 = 27
Second case we’ll calculate in ascending order to evade double counting:
3*5=15
3*7=21
3*11=33
….
3*31=93
____________
5*7=35
5*11=55

5*19=95
____________
7*11=77
7*13=91
Total 16

Total number of integers = 16+1=17

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Joined: 12 Aug 2015
Posts: 2621
Schools: Boston U '20 (M)
GRE 1: Q169 V154
How many integers less than 100 have exactly 4 odd factors b  [#permalink]

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16 Jan 2017, 04:44
Great Question.
Here is what i did on this one ->
Number can be of the form -> Prime^3 or Prime*prime
Where no prime is 2.

Hence the numbers will be ->
3^3
3*5
3*7
3*11
3*13
3*17
3*19
3*23
3*29
3*31
5*7
5*11
5*13
5*17
5*19
7*11
7*13
THAT'S IT.
Clearly there are 17 such possibilities.
Hence D.

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Joined: 22 May 2015
Posts: 108
Re: How many integers less than 100 have exactly 4 odd factors b  [#permalink]

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16 Jan 2017, 08:45
The question is easy to solve if we know this rule , no of odd factors for a number X = a^p * b^q where a and b are prime numbers is (p+1)*(q+1)

So possible ways to have 4 odd factors are X = a*b or a^3.
Listing the prime numbers 3,5,7,11,13,17,19,23,29,31
for 3 - 9 combinations+1 combination ( 3^3)
for 5 - 5 combinations
for 7 - 2 combinations
=>17
_________________

Consistency is the Key

Intern
Joined: 10 Dec 2016
Posts: 17
Re: How many integers less than 100 have exactly 4 odd factors b  [#permalink]

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06 Feb 2019, 06:22
Hi Bunuel ,

Can we consider negative integers in this case?
Re: How many integers less than 100 have exactly 4 odd factors b   [#permalink] 06 Feb 2019, 06:22

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