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How many two digit integers have exactly five divisors?

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guerrero25
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How many two digit integers have exactly five divisors? [#permalink] New post   28 Aug 2014, 11:58
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How many two digit integers have exactly five divisors?

A)Zero

B)One

C)Two

D)Three

E)Four
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C
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Bunuel
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Re: How many two digit integers have exactly five divisors? [#permalink] New post   28 Aug 2014, 12:09
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guerrero25 wrote:
How many two digit integers have exactly five divisors?

A)Zero

B)One

C)Two

D)Three

E)Four


Finding the Number of Factors of an Integer

First make prime factorization of an integer \(n=a^p*b^q*c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\) and \(p\), \(q\), and \(r\) are their powers.

The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: \(450=2^1*3^2*5^2\)

Total number of factors of 450 including 1 and 450 itself is \((1+1)*(2+1)*(2+1)=2*3*3=18\) factors.

Back to the question:

Since 5 is a prime number, it cannot be the product of two integers greater than 1, which implies that a number having 5 factors must be of a form of (prime)^4 --> the number of factors = (4 + 1).

There are only 2 two-digit numbers which can be written this way: 2^4 = 16 and 3^4 = 81.

Answer: C.
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How many two digit integers have exactly five divisors? [#permalink] New post   Updated on: 17 Oct 2022, 01:24
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guerrero25 wrote:
How many two digit integers have exactly five divisors?

A)Zero

B)One

C)Two

D)Three

E)Four


Note that if we generalize this question, it could become cumbersome - i.e. if we change it to "How many two digit integers have exactly four divisors?", it will involve quite a bit of work. There is something special about 5 divisors which makes it easy to solve if you understand the fundamentals.
If a number has 3 factors, it means it is a perfect square. If it has 5 factors, it means it is a power of 4 of a prime number. If it has 7 factors, it means it is a power of 6 of a prime number. If it has 9 factors, it means it is either a power of 8 of a prime number or has squares of 2 prime numbers.
You must understand why each of these is true. To do so, check out Bunuel's explanation above.

Here, since the number has 5 factors, it must be a power of 4 of a prime number. The fourth power of 2 is 16 (a two digit number). The fourth power of 3 is 81 (a two digit number). The fourth power of 5 is 625 (a three digit number so not acceptable). All other prime numbers will have fourth power higher than 625 so ignore them.

So the number could be either 16 or 81 i.e. two values.

Answer (C)

Originally posted by KarishmaB on 19 Jan 2015, 22:47.
Last edited by KarishmaB on 17 Oct 2022, 01:24, edited 1 time in total.
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Re: How many two digit integers have exactly five divisors? [#permalink] New post   28 Aug 2014, 12:18
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Bunuel wrote:
guerrero25 wrote:
How many two digit integers have exactly five divisors?

A)Zero

B)One

C)Two

D)Three

E)Four


Finding the Number of Factors of an Integer

First make prime factorization of an integer \(n=a^p*b^q*c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\) and \(p\), \(q\), and \(r\) are their powers.

The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: \(450=2^1*3^2*5^2\)

Total number of factors of 450 including 1 and 450 itself is \((1+1)*(2+1)*(2+1)=2*3*3=18\) factors.

Back to the question:

Since 5 is a prime number, it cannot be the product of two integers greater than 1, which implies that a number having 5 factors must be of a form of (prime)^4 --> the number of factors = (4 + 1).

There are only 2 two-digit numbers which can be written this way: 2^4 = 16 and 3^4 = 81.

Answer: C.


Similar questions to practice:
if-q-a-positive-integer-has-5-factors-which-of-the-foll-149058.html
how-many-integers-less-than-100-have-exactly-4-odd-factors-b-97404.html
if-n-has-15-positive-divisors-inclusive-of-1-and-n-then-wh-154437.html
if-the-integer-n-has-exactly-three-positive-divisors-includ-145985.html
if-a-natural-number-p-has-8-factors-then-which-of-the-142265.html
if-x-is-an-integer-that-has-exactly-three-positive-divisors-132710.html
find-the-number-of-factors-of-a-three-digit-even-number-xyz-128546.html
if-the-prime-factorization-of-the-integer-q-can-be-expressed-126689.html
an-even-positive-integer-x-has-y-positive-integral-facto-107733.html

Hope this helps.
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Re: How many two digit integers have exactly five divisors? [#permalink] New post   22 Jul 2015, 01:45
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nvad5955 wrote:
Bunuel wrote:
guerrero25 wrote:
How many two digit integers have exactly five divisors?

A)Zero

B)One

C)Two

D)Three

E)Four


Finding the Number of Factors of an Integer

First make prime factorization of an integer \(n=a^p*b^q*c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\) and \(p\), \(q\), and \(r\) are their powers.

The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: \(450=2^1*3^2*5^2\)

Total number of factors of 450 including 1 and 450 itself is \((1+1)*(2+1)*(2+1)=2*3*3=18\) factors.

Back to the question:

Since 5 is a prime number, it cannot be the product of two integers greater than 1, which implies that a number having 5 factors must be of a form of (prime)^4 --> the number of factors = (4 + 1).

There are only 2 two-digit numbers which can be written this way: 2^4 = 16 and 3^4 = 81.

Answer: C.



shouldn't the answer be 4?: 32,81,-32,-81
-32 is a 2 digit number with 5 divisors -1,-2,-4,-8,-32
the same with -81.


When we talk about factors/divisors, it is assumed that we are talking about positive integers only.
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Re: How many two digit integers have exactly five divisors? [#permalink] New post   06 Apr 2016, 18:43
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only perfect squares have odd numbers of factors.
and only perfect squares of 4, 5, 6, 7, 8, and 9 are two digit.
so: 16, 25, 36, 49, 64, 81
now..16=2^4. 4+1 = 5 factors, so works.
25 - 5^2 -> 3 factors, out.
36 = 2^2 * 3^2 = 3x3=9 factors, out.
49 = 7^2 = 3 factors, out.
64 = 2^6 -> 7 factors, out.
81 = 3^4 => 5 factors.

we have 2 two digit numbers with 5 factors.
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Re: How many two digit integers have exactly five divisors? [#permalink] New post   28 Feb 2018, 11:11
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guerrero25 wrote:
How many two digit integers have exactly five divisors?

A)Zero

B)One

C)Two

D)Three

E)Four


The only numbers that have exactly five divisors are those can be expressed in the form of p^4 where p is a prime (recall that we can add 1 to the exponent to obtain the number of divisors of a number).

Since 2^4 = 16 and 3^4 = 81 (and 5^4 = 625 would not be a two-digit number), there are 2 two-digit numbers that have exactly five divisors.

Answer: C
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How many two digit integers have exactly five divisors? [#permalink] New post   19 Jan 2015, 20:40
Quote:

Similar questions to practice:
if-q-a-positive-integer-has-5-factors-which-of-the-foll-149058.html
how-many-integers-less-than-100-have-exactly-4-odd-factors-b-97404.html
if-n-has-15-positive-divisors-inclusive-of-1-and-n-then-wh-154437.html
if-the-integer-n-has-exactly-three-positive-divisors-includ-145985.html
if-a-natural-number-p-has-8-factors-then-which-of-the-142265.html
if-x-is-an-integer-that-has-exactly-three-positive-divisors-132710.html
find-the-number-of-factors-of-a-three-digit-even-number-xyz-128546.html
if-the-prime-factorization-of-the-integer-q-can-be-expressed-126689.html
an-even-positive-integer-x-has-y-positive-integral-facto-107733.html



Yet another one:

Question: How many factors does the integer 9999 have?

Show SpoilerAnswer
99999 is the same as 10^4 - 1. This allows us to use the difference of squares to our advantage as follows:

(100+1) (100-1) = 101 * 99.

With this we know that 101 is prime, and 99 can be expressed as (3)(3)(11), which will result in the following factorization: 3*3*11*101 = (3^2)*11*101.

using the explanation in the first post of the thread we can get to the answer of 12 unique factors.
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Re: How many two digit integers have exactly five divisors? [#permalink] New post   22 Jul 2015, 00:35
Bunuel wrote:
guerrero25 wrote:
How many two digit integers have exactly five divisors?

A)Zero

B)One

C)Two

D)Three

E)Four


Finding the Number of Factors of an Integer

First make prime factorization of an integer \(n=a^p*b^q*c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\) and \(p\), \(q\), and \(r\) are their powers.

The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: \(450=2^1*3^2*5^2\)

Total number of factors of 450 including 1 and 450 itself is \((1+1)*(2+1)*(2+1)=2*3*3=18\) factors.

Back to the question:

Since 5 is a prime number, it cannot be the product of two integers greater than 1, which implies that a number having 5 factors must be of a form of (prime)^4 --> the number of factors = (4 + 1).

There are only 2 two-digit numbers which can be written this way: 2^4 = 16 and 3^4 = 81.

Answer: C.



shouldn't the answer be 4?: 32,81,-32,-81
-32 is a 2 digit number with 5 divisors -1,-2,-4,-8,-32
the same with -81.
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Re: How many two digit integers have exactly five divisors? [#permalink] New post   23 Jan 2016, 04:33
the original questions reads "positive divisors"
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Re: How many two digit integers have exactly five divisors? [#permalink] New post   20 Dec 2016, 07:20
mvictor wrote:
only perfect squares have odd numbers of factors.
and only perfect squares of 4, 5, 6, 7, 8, and 9 are two digit.
so: 16, 25, 36, 49, 64, 81
now..16=2^4. 4+1 = 5 factors, so works.
25 - 5^2 -> 3 factors, out.
36 = 2^2 * 3^2 = 3x3=9 factors, out.
49 = 7^2 = 3 factors, out.
64 = 2^6 -> 7 factors, out.
81 = 3^4 => 5 factors.

we have 2 two digit numbers with 5 factors.


I did it in exactly the same way. I know it is application of brute force but it atleast helped me in getting the correct answer within the time frame.
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Re: How many two digit integers have exactly five divisors? [#permalink] New post   19 Feb 2017, 07:37
VeritasPrepKarishma

Can you please share an instance from the official material that supports the same? It seems to me that +2 should be considered a divisor of -32.
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Re: How many two digit integers have exactly five divisors? [#permalink] New post   26 Feb 2018, 12:04
Well Only perfect squares have odd no of factors or divisors..'

SO if you consider no of factors for all the squares from 1-9, Only 16 and 81 falls under the required category..

Hence Ans is 2.
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Re: How many two digit integers have exactly five divisors? [#permalink] New post   16 May 2018, 22:09
ScottTargetTestPrep wrote:
guerrero25 wrote:
How many two digit integers have exactly five divisors?

A)Zero

B)One

C)Two

D)Three

E)Four


The only numbers that have exactly five divisors are those can be expressed in the form of p^4 where p is a prime (recall that we can add 1 to the exponent to obtain the number of divisors of a number).

Since 2^4 = 16 and 3^4 = 81 (and 5^4 = 625 would not be a two-digit number), there are 2 two-digit numbers that have exactly five divisors.

Answer: C


Another way to approach the problem is by means of fact that "Perfect square will only have odd number of factors". Now in the range 10-99,
following are the perfect square : 16,25,36,64,81.Out of which only 16 and 81 have powers of 4 which means number of factors is 4+1 =5.

Hence the answer C.
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Re: How many two digit integers have exactly five divisors? [#permalink] New post   20 Sep 2020, 23:23
In order for a Number to have 5 (+)Positive Divisors, it can only take the following Form:


N = (p)^4th Power


**where p = Prime Base


Possible 2 Digit Nos. that fit this Form:

N = (2)^4 = 16

N = (3)^4 = 81

N = (4)^4 = too large


only TWO 2 Digit Numbers are possible with 5 Divisors

-C-
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Re: How many two digit integers have exactly five divisors? [#permalink] New post   20 Sep 2020, 23:23
In order for a Number to have 5 (+)Positive Divisors, it can only take the following Form:


N = (p)^4th Power


**where p = Prime Base


Possible 2 Digit Nos. that fit this Form:

N = (2)^4 = 16

N = (3)^4 = 81

N = (4)^4 = too large


only TWO 2 Digit Numbers are possible with 5 Divisors

-C-
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Re: How many two digit integers have exactly five divisors? [#permalink] New post   23 Dec 2020, 13:17
Bunuel, is this GMAT material? Should I learn number of unique divisors?
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Re: How many two digit integers have exactly five divisors? [#permalink] New post   30 Jan 2022, 08:06
OE:
Following the process for finding the number of unique divisors, we realize that an integer with exactly five divisors MUST be the fourth power of a prime number. Hence 16 and 81 are the only options.

Another way to look at this problem - the only numbers with an odd number of divisors are perfect squares (for example, 9 has three divisors: 1, 3, and 9). Since this problem asks about two-digit numbers, trial-and-error can help as there are only 16, 25, 36, 49, 64, and 81 to try to factor.
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