7. Five cards have the numbers $101,102,103,104$ and 105 on their fronts.
101
102
103
104
105
On the reverse, each card has one of five different positive integers: $a, b, c, d$ and $e$ respectively.
We know that $c=b e, a+b=d$ and $e-d=a$.
Frankie picks up the card which has the largest integer on its reverse. What number is on the front of Frankie's card?
SOLUTION
103

Since $e-d=a$ we have $e=a+d$. Therefore, $e>a$ and $e>d$.
Since $c=b e$ we have $c>e$ and $c>b$, for if either of $e, b$ equalled 1 then $c$ would equal the other.
Therefore, $c>e>a, c>e>d$ and $c>b$.
The largest integer is $c$ and the correct answer is 103 .