AMC10 2006 10B, Problem 22

Elmo makes $N$ sandwiches for a fundraiser. For each sandwich he uses $B$ globs of peanut butter at $4\cent$ per glob and $J$ blobs of jam at $5\cent$ per blob. The cost of the peanut butter and jam to make all the sandwiches is $$ 2.53$. Assume that $B$, $J$, and $N$ are positive integers with $N>1$. What is the cost of the jam Elmo uses to make the sandwiches?

Solution: The peanut butter and jam for each sandwich costs $4B\cent+5J\cent$, so the peanut butter and jam for $N$ sandwiches costs $N(4B+5J)\cent$. Setting this equal to $253\cent$: $N(4B+5J)=253=11\cdot23$ The only possible positive integer pairs $(N , 4B+5J)$ whose product is $253$ are: $(1,253) ; (11,23) ; (23,11) ; (253,1)$ The first pair violates $N>1$ and the third and fourth pairs have no positive integer solutions for $B$ and $J$. So, $N=11$ and $4B+5J=23$. The only integer solutions for $B$ and $J$ are $B=2$ and $J=3$. Therefore, the cost of the jam Elmo uses to make the sandwiches is $3\cdot5\cdot11=165\cent$  $=  $1.65 \Rightarrow \fbox{1.65}$

Answer: 1.65