In Yacleyland, the unit of money used is called the Yacley. There are only two denominations of paper money: the 17 Yacley bill and the 5 Yacley bill. How many different combinations of these bills total 453 Yacley?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7
We start by determining the combination of these bills totalling 453 Yacleys which uses the fewest 17 Yacley bills.

To do this, we notice that since we can use as many 5 Yacley bills as we'd like, then any multiple of 17 less than 453 which ends in a 5 or an 8 can be "topped up" to 453 Yacleys using 5 Yacley bills.

The first few multiples of 17 are 17, 34, 51, 68 .

So if we use four 17 Yacley bills, we have 68 Yacleys, leaving $453-68=385$ Yacleys to get to 453 . For 385 Yacleys, we need 77 of the 5 Yacley bills.

So 4 of the 17 Yacley bills and 77 of the 5 Yacley bills works.

To get other combinations, we use the fact that 5 of the 17 Yacley bills are worth the same as 17 of the 5 Yacley bills, so we can subtract 17 of the 5 Yacley bills and add 5 of the 17 Yacley bills and keep the total the same.

Thus, $77-17=60$ of the 5 Yacley bills and $4+5=9$ of the 17 Yacley bills make 453 Yacleys. (Check this!)

Also, 43 and 14 of the two types of bills, and 26 and 19 of the two types of bills, and 9 and 24 of the two types of bills.

Since we are down to 9 of the 5 Yacley bills, we can no longer use this exchanging process (since we need at least 17 of the 5 Yacley bills to be able to do this).

Therefore, there are 5 combinations that work.

(An alternate approach would be to let $x$ be the number of 17 Yacley bills and $y$ the number of 5 Yacley bills used.

We then would like to consider the equation $17 x+5 y=453$, and find the number of pairs $(x, y)$ which satisfy this equation and where both $x$ and $y$ are positive integers.

Geometrically, we are trying to find the number of points $(x, y)$, with $x$ and $y$ both positive integers, lying on the line $17 x+5 y=453$.

In a similar way to above, we can find that $(x, y)=(4,77)$ is such a point.

Since the slope of the line is $-\frac{17}{5}$, then to get a another point with integer coordinates on the line, we can move 5 units right and 17 units down.

![](https://cdn.mathpix.com/cropped/2024_04_20_688a3b42f69739f2e244g-207.jpg?height=439&width=445&top_left_y=564&top_left_x=878)

We can repeat this process as above, and get 5 combinations that work.)

ANSWER: (C)