To make the mathematical statement true, we need to find the value of \( x \) such that:
\[ (17 + p) + 15 = (17 + x) + p \]
We can simplify both sides.
The left-hand side becomes: \[ (17 + p) + 15 = 17 + p + 15 = 32 + p \]
The right-hand side becomes: \[ (17 + x) + p = 17 + x + p \]
Now we can set the two expressions equal to one another: \[ 32 + p = 17 + x + p \]
We can subtract \( p \) from both sides: \[ 32 = 17 + x \]
Now, solve for \( x \) by subtracting \( 17 \) from both sides: \[ 32 - 17 = x \] \[ 15 = x \]
So, the number that makes the statement true is \( \boxed{15} \).