1. ## Divisibility

I know I am supposed to use Divisibility Theorems to solve this problem, but I am not sure how to go about doing it.

An old receipt has faded. It reads 88 chickens at a total cost of x4.2y$where x and y are unreadable. How much did each chicken cost? 2. Originally Posted by RonyPony I know I am supposed to use Divisibility Theorems to solve this problem, but I am not sure how to go about doing it. An old receipt has faded. It reads 88 chickens at a total cost of x4.2y$ where x and y are unreadable. How much did each chicken cost?
Assume a chicken costs an integer number of cents.

Then x42y is divisible by 8 and 11. y must be even, and we can use the test for divisibility by 11 using digits: x - 4 + 2 - y is divisible by 11. That is, x - y - 2 is divisible by 11. It can be seen that 0 is the only attainable number, so x-y-2=0 and we try

(x,y) = (2,0)
(x,y) = (4,2)
(x,y) = (6,4)
(x,y) = (8,6)

The only one that works is (x,y) = (6,4), so we have $64.24 and each chicken costs 73 cents. 3. Originally Posted by undefined That is, x - y - 2 is divisible by 11. It can be seen that 0 is the only attainable number, so x-y-2=0 and we try I'm not sure what you mean by that or how you got it, can you explain it in a little more detail. 4. Originally Posted by RonyPony I'm not sure what you mean by that or how you got it, can you explain it in a little more detail. Divisibility Tests 5. Hello, RonyPony! A slight variation of undefined's solution. An old receipt has faded. It reads 88 chickens at a total cost of .$\displaystyle \$x4.2y$, where $\displaystyle x$ and $\displaystyle y$ are unreadable.
How much did each chicken cost?

88 chickens cost $\displaystyle N \,=\, x42y$ cents.
Hence, $\displaystyle N$ is divisible by 8 and 11.

A number is divisible by 8 if its last three-digit number is divisible by 8.
. . That is: .$\displaystyle 42y$ must be divisible by 8.
Since $\displaystyle y$ is a digit, the only choice is: .$\displaystyle y \,=\,4$
. . And we have: .$\displaystyle N \,=\,x424$

A number is divisible by 11 if: .$\displaystyle \text{(sum of digits in "odd" positions)}$
. . $\displaystyle \text{minus}\:\text{(sum of digits in "even" positions)}\; =\; \text{(multiple of 11)}$
That is: .$\displaystyle (x+2) - (4+4) \:=\:11a$
. . and we have: .$\displaystyle x \:=\:11a + 6$
Since $\displaystyle x$ is a digit: .$\displaystyle a = 0 \quad\Rightarrow\quad x \,=\,6$

Hence: .$\displaystyle N \:=\:6424$

Each chicken cost: .$\displaystyle 6424 \div 88 \:=\:73\text{ cents.}$