27. At a fun fair, coupons can be used to purchsse food. Each coupon is worth $\$ 5, \$ 8$ or $\$ 12$. For example, for a $\$ 15$ purchase you can use three coupons of $\$ 5$, or use one coupon of $\$ 5$ and one coupon of $\$ 8$ and pay $\$ 2$ by cash. Suppose the prices in the fun fair are all whole dollars. What is the largest amount that you cannot purchase using only coupons?
27. Answer: 19 .
Note that $8+12=20,5+8+8=21,5+5+12=22,5+5+5+8=23,8+8+8=24$. If $n \geq 25$, write $n=5 k+m$ where $20 \leq m \leq 24$ and $k$ is a positive integer. So any amount $\geq 25$ can be paid exactly using coupons.
However, 19 cannot be paid exactly using these three types of coupons.