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

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How many numbers between 1 and 100, inclusive, have exactly 5 positive [#permalink]

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New post 18 Apr 2017, 04:53
1
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A
B
C
D
E

Difficulty:

  55% (hard)

Question Stats:

57% (00:52) correct 43% (01:03) wrong based on 219 sessions

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Re: How many numbers between 1 and 100, inclusive, have exactly 5 positive [#permalink]

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New post 18 Apr 2017, 07:23
2
Bunuel wrote:
How many numbers between 1 and 100, inclusive, have exactly 5 positive factors?

A. 0
B. 1
C. 2
D. 9
E. 10


Just giving this question a try.

5 positive factors are possible if a^4 (a= prime no. and 4 is its power i.e. a perfect square).
NOTE: Perfect squares have odd number of total factors.

Total no. of factors= a^(4+1)= a^5

There can be only one prime no. as 5 is a prime no. and has no factors that after being multiplied would give 5.

So the possible values between 1 and 100 inclusive that have exactly 5 positive factors are 2^4 i.e. (16) and 3^4 i.e.(81).

Only two values as 4^4= 256 which is greater than 100.
Also I did not take 1 as it has only one factor 1.

So I chose C as the option.
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Re: How many numbers between 1 and 100, inclusive, have exactly 5 positive [#permalink]

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New post 18 Apr 2017, 08:52
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Bunuel 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:

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Re: How many numbers between 1 and 100, inclusive, have exactly 5 positive [#permalink]

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New post 24 Apr 2017, 16:53
1
4
Bunuel wrote:
How many numbers between 1 and 100, inclusive, have exactly 5 positive factors?

A. 0
B. 1
C. 2
D. 9
E. 10


We should recall the rule we can use to determine the total number of factors:

For a positive integer n (where n > 1),
i) if the prime factorization of n is p^a (where p is a prime), then the total number of factors of n is equal to a + 1.
ii) if the prime factorization of n is p^a * q^b (where p and q are distinct primes), then the total number of factors of n is equal to (a + 1)(b + 1).
(Note: The concept can be extended to the prime factorization of n when n has 3 or more distinct prime factors.)

Since 5 is a prime number, the number(s) we seek can’t have more than 1 prime factor. For example, if it has two distinct prime factors, then, according to the rule, the total number of factors is equal to (a + 1)(b + 1). But (a + 1)(b + 1) can’t be equal to 5 since 5 is a prime number..

Thus we see that the number must have only 1 prime factor and is of the form p^a. Since, according to the rule, the total number of factors is equal to a + 1, we can set a + 1 = 5 and obtain a = 4. Now let’s check some values of p (keep in mind that p is a prime):

If p = 2, then p^4 = 2^4 = 16 (which is between 1 and 100)
If p = 3, then p^4 = 3^4 = 81 (which is between 1 and 100)
If p = 5, then p^4 = 5^4 = 625 (which is more than 100)

Thus, there are only two integers between 1 and 100 (6 and 81) that have exactly 5 positive factors.

Answer: C
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Re: How many numbers between 1 and 100, inclusive, have exactly 5 positive [#permalink]

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New post 25 Apr 2017, 13:15
Bunuel wrote:
How many numbers between 1 and 100, inclusive, have exactly 5 positive factors?

A. 0
B. 1
C. 2
D. 9
E. 10


Integers with an odd number of factors are square. Listing the squares from 1-100 inclusive, shows you that only 16 and 81 have 5 factors.
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Re: How many numbers between 1 and 100, inclusive, have exactly 5 positive [#permalink]

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New post 01 May 2017, 14:31
I got this question wrong even though I knew the rule that perfect squares have an odd number of factors. I started testing numbers at 25, working up to 100, so I missed 16. Doh!!
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New post 01 May 2017, 20:52
1
The no should have 5 factors so it must be in in the form of
5*1
(4+1) (0+1)
a^4

a should be a prime no and only two value satisfy the given condition .ie 2 and 3

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Re: How many numbers between 1 and 100, inclusive, have exactly 5 positive [#permalink]

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Re: How many numbers between 1 and 100, inclusive, have exactly 5 positive   [#permalink] 09 May 2018, 05:28
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