How many two digit integers have exactly five divisors?

Question

A)Zero

B)One

C)Two

D)Three

E)Four

Bunuel · Accepted Answer

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.

KarishmaB · Answer

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)

Bunuel · Answer

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msad · Answer

Yet another one:

Question: How many factors does the integer 9999 have?

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.

nvad5955 · Answer

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.

KarishmaB · Answer

When we talk about factors/divisors, it is assumed that we are talking about positive integers only.

mvictor · Answer

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.

meetsampat16 · Answer

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.

ScottTargetTestPrep · Answer

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

nayanparikh · Answer

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.

Fdambro294 · Answer

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-

rvgmat12 · Answer

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.

Archit3110 · Answer

prime number raised to power 4 will have 5 divisors
2 digit numbers possible 16 , 81
2^4 & 3^4
option C

ashKing12 · Answer

so, number of divisors / factors = 5 
All perfect square numbers have ODD number of factors
2-digit perfect squares are 16,25,36,49,64,81

square of a prime has exactly 3 factors so 25 and 49 can't have 5 factors

Find the total number of factors for 16, 36, 64, 81

Only 16 and 81 has exactly 5 factors (so C)

How many two digit integers have exactly five divisors?

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