+26404 I am less than 500. All three digits are odd. All three digits are different. The sum of the digits is thirteen. The product of my digits is greater than 30. I am not divisible by 5. Who am I?
\(\begin{array}{lrcll} (1) & x + y + z &=& 13 \\ & z &=& 13-(x+y) \\ (2) & x\cdot 100 + y\cdot 10 + z &<& 500 \\ & x\cdot 100 + y\cdot 10 + 13-(x+y) &<& 500 \\ & 99x + 9y + 13 &<& 500 \\ & 9y &<& - 99x + 500 - 13 \\ & 9y &<& - 99x + 487 \\ && & \boxed{~ \mathbf{ y } ~ \mathbf{<}~ \mathbf{- 11x + \frac{487}{9} } \\ ~}\\ (3) & x\cdot y \cdot z &>& 30\\ & x\cdot y &>& \frac{30}{z}\\ &&& \boxed{~ \mathbf{x\cdot y } ~\mathbf{>}~ \mathbf{\frac{30}{13-(x+y)} }\\ ~} \end{array} \)
I have put \(y ~<~ - 11x + \frac{487}{9} \) and \(x\cdot y ~>~ \frac{30}{13-(x+y)} \) in https://www.desmos.com/calculator
se we have a result of 5 Points. see: 
Now let us see:
\(\begin{array}{|r|r|r|r|} \hline x & y & z=13-(x+y) & \text{result}\\ \hline 1 & 5 & 13 - (1+5) = 7 & {\color{red}\text{yes}}\\ 1 & 7 & 13 - (1+7) = 5 & \text{no, because }~ 175 ~\text{ is divisible by }~ 5\\ 3 & 3 & 13 - (3+3) = 7 & \text{no, because }~ x = y \\ 3 & 5 & 13 - (3+5 )= 5 & \text{no, because }~ y = z \\ 3 & 7 & 13 - (3+7 )= 3 & \text{no, because }~ x = z \\ \hline \end{array} \)
There is only one number 157
