# Divisibility

• Jul 6th 2010, 10:39 AM
RonyPony
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$where x and y are unreadable. How much did each chicken cost? • Jul 6th 2010, 10:51 AM undefined Quote: Originally Posted by RonyPony I know I am supposed to use Divisibility Theorems to solve this problem, but I am not sure how to go about doing it. An old receipt has faded. It reads 88 chickens at a total cost of x4.2y$ 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.
• Jul 6th 2010, 12:26 PM
RonyPony
Quote:

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.
• Jul 6th 2010, 12:58 PM
undefined
Quote:

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
• Jul 6th 2010, 02:39 PM
Soroban
Hello, RonyPony!

A slight variation of undefined's solution.

Quote:

It reads 88 chickens at a total cost of . $\x4.2y$, where $x$ and $y$ are unreadable.
How much did each chicken cost?

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

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

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

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

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