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How many numbers between 1 and 100, inclusive, have exactly 5 positive
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28 Jun 2021, 07:00
28 Jun 2021, 07:00
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63% (01:22) correct
36% (00:50) wrong based on 22 sessions
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How many numbers between 1 and 100, inclusive, have exactly 5 positive factors?
A. 0
B. 1
C. 2
D. 9
E. 10
A. 0
B. 1
C. 2
D. 9
E. 10
Re: How many numbers between 1 and 100, inclusive, have exactly 5 positive
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28 Jun 2021, 07:51
28 Jun 2021, 07:51
1
Solution:
Finding every number's factor will be a tedious.
Instead, we can try multiplying the numbers.
What I mean is, when a number has 5 positive factors, it means it is multiplied 4 times.
\(2^4\)=32
\(3^4\)=81
\(4^4\)=256
Thus, only 2 & 3 satisfy the above condition.
IMO C
Hope this helps!
Finding every number's factor will be a tedious.
Instead, we can try multiplying the numbers.
What I mean is, when a number has 5 positive factors, it means it is multiplied 4 times.
\(2^4\)=32
\(3^4\)=81
\(4^4\)=256
Thus, only 2 & 3 satisfy the above condition.
IMO C
Hope this helps!
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Posts: 6218
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Re: How many numbers between 1 and 100, inclusive, have exactly 5 positive
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28 Jun 2021, 07:54
28 Jun 2021, 07:54
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Carcass wrote:
How many numbers between 1 and 100, inclusive, have exactly 5 positive factors?
A. 0
B. 1
C. 2
D. 9
E. 10
A. 0
B. 1
C. 2
D. 9
E. 10
First note that most positive integers have an EVEN number of positive factors.
Only the SQUARES of integers have an ODD number of positive factors.
So, we need only consider the following squares of integers: 1, 4, 9, 16, 25, 36, . . . . 81, 100
Most of these squares don't have 5 positive factors. Let's check...
Factors of 1: 1 NO
Factors of 4: 1, 2, 4 NO
Factors of 9: 1, 3, 9 NO
Factors of 16: 1, 2, 4, 8, 16 PERFECT!
Factors of 25: 1, 5, 25 NO
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 NO
Factors of 49: 1, 7, 49 NO
Factors of 64: 1, 2, 4, 8, 16, 32, 64 NO
Factors of 81: 1, 3, 9, 27, 81 PERFECT!
Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100 NO
Answer: C
