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How many numbers between 1 and 100, inclusive, have exactly 5 positive

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Carcass
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How many numbers between 1 and 100, inclusive, have exactly 5 positive [#permalink] New post   28 Jun 2021, 07:00
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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
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C
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Re: How many numbers between 1 and 100, inclusive, have exactly 5 positive [#permalink] New post   28 Jun 2021, 07:51
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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!
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Re: How many numbers between 1 and 100, inclusive, have exactly 5 positive [#permalink] New post   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


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
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Re: How many numbers between 1 and 100, inclusive, have exactly 5 positive [#permalink] New post   28 Dec 2021, 11:02
To find the number of factors a number has, we carry out prime factorisation, add one to the exponent of each of the prime factors and then multiply those figures. Against this background, we can only obtain 5 if there is only one prime factor and if that prime factor has 4 as its exponent. So, our focus will be on prime numbers that are raised to the power of 4 and which give numbers that are between the specified 1 to 100 range. Only 2 and 3 raised to the power of 4 satisfy this condition, giving numbers 16 and 81 respectively. The answer is thus option C.

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Re: How many numbers between 1 and 100, inclusive, have exactly 5 positive [#permalink]
28 Dec 2021, 11:02
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