SOLUTION: There is a positive two-digit integer that is divisible by the sum of its digits AND that has exactly 5 factors. What is the sum of its factors?

Question 491972: There is a positive two-digit integer that is divisible by the sum of its digits AND that has exactly 5 factors. What is the sum of its factors?
Answer by Edwin McCravy(20060) (Show Source): You can put this solution on YOUR website!

For any integer the numbers

A. one of its factors 

and 

B.  

are two different integers except when the integer is a
perfect square and that factor is its square root.

So only perfect squares have an odd number of factors, since
its square root and  are not two different
factors, but only one factor, since those are the same number. 

So it has to be a perfect square to have 5 factors.
So it has to be one of these:

16, 25, 36, 49, 64, or 81

16 is not divisible by 1+6, which is 7

25 is not divisible by 2+5, which is 7

36 IS divisible by 3+6, which is 9.  However 36 has 
       9 factors 1,2,3,4,6,9,12,18, and 36, and that's too many.

64 is not divisible by 6+4, which is 10.

81 is divisible by 8+1, which is 9.  It's factors are 1,3,9,27, and 81.
       And that's 5, so the number is 81.

Answer: The sum of the factors is 1+3+9+27+81 = 121

Edwin

RELATED QUESTIONS
There is only one positive integer that is exactly twice the sum of its digits. What... (answered by dabanfield,oberobic)

a two digit number has each of the digits {0, 1, 2, 3, 4, 5, 6} appearing exactly once.... (answered by math_helper)

find a number that; The last digit is even Sum of the digits is divisible by 3 The... (answered by jim_thompson5910)

find the least 5 digit number which is exactly divisible by 20,25,30? and find the... (answered by Edwin McCravy)

a three-digit number, which is divisible by 10, has a hundreds digit that is ove less... (answered by stanbon,gonzo)

A three-digit number, which is divisible by 10, has a hundreds digit that is one less... (answered by dabanfield)

A three digit number divisible by ten has a hundreds digit that is one less than its tens (answered by ankor@dixie-net.com)

There is a two digit number that is 9 times the sum of its digits. What is this... (answered by ankor@dixie-net.com)

Choose one number at random from 1, 2, 3, 4, ... , 999, 1000. What is the probability... (answered by edjones)