SOLUTION: PROBLEM 1: A two digit number with 2 different digits has a special property: "When the sum of its digits is added to the product of its digits, the result is the number itself."

Question 431642: PROBLEM 1:
A two digit number with 2 different digits has a special property: "When the sum of its digits is added to the product of its digits, the result is the number itself." What is the smallest number with this property?
PROBLEM 2:
Six is a perfect number because its factors ( not including 6)add up to itself. What are all the perfect numbers between 20 and 30?
Found 2 solutions by ankor@dixie-net.com, richard1234:
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
A two digit number with 2 different digits has a special property:
"When the sum of its digits is added to the product of its digits, the result is the number itself."
What is the smallest number with this property?
:
x = 10's digit
y = units
then
10x+y = the number
:
x + y + xy = 10x + y
xy = 10x - x + y - y
xy = 9x
y =
y = 9 is the units
then
19 is the smallest number with this property.
:
See if that is true
1 + 9 + (1*9) = 10(1) + 9
:
:
PROBLEM 2:
Six is a perfect number because its factors ( not including 6)add up to itself. What are all the perfect numbers between 20 and 30?
28: 1 + 2 + 4 + 7 + 14

Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
1. Suppose the number is , where . Then, --> --> , b = 9. We can assume to be as small as possible, so 19 is the smallest such number.

2. 28, because the sum of its proper divisors is 1+2+4+7+14 = 28. In fact, all even perfect numbers are in the form where is prime. This is because the sum of divisors function is a multiplicative function for relatively prime integers. When , , prime, so , a perfect number.
RELATED QUESTIONS
A 2 digit number with 2 different digits has a special property: " when the sum of its... (answered by Edwin McCravy)
The sum of the two digits of a two-digit number is 6. When 18 is added to the number... (answered by Edwin McCravy)
A two digit number is 4 times the sum of its digits. If 27 is added to that number its... (answered by josmiceli)
A two-digit number is such that the product of its digits is 12.When 36 is added to this... (answered by longjonsilver)
The sum of the digits of a 2-digit number is 8. When 36 is added to the number, the... (answered by ankor@dixie-net.com)
[ A two digit number is three times the sum of its digits. if 45 is added to the number,... (answered by TimothyLamb)
A two-digit number is eight times the sum of its digit. When the number is added to the... (answered by ankor@dixie-net.com)
The sum of the digits of a two-digit number is 7. When 14 is added to the original... (answered by lwsshak3)
The sum of the digits of a two-digit number is 8. If 16 is added to the original number, (answered by ankor@dixie-net.com)