Results 1 to 5 of 5

Math Help - Divisibility

  1. #1
    Newbie
    Joined
    Jun 2010
    Posts
    11

    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?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor undefined's Avatar
    Joined
    Mar 2010
    From
    Chicago
    Posts
    2,340
    Awards
    1
    Quote Originally Posted by RonyPony View Post
    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.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jun 2010
    Posts
    11
    Quote Originally Posted by undefined View Post
    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.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor undefined's Avatar
    Joined
    Mar 2010
    From
    Chicago
    Posts
    2,340
    Awards
    1
    Quote Originally Posted by RonyPony View Post
    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
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,748
    Thanks
    648
    Hello, RonyPony!

    A slight variation of undefined's solution.


    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?

    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.}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Divisibility 11
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: December 20th 2008, 02:41 AM
  2. Divisibility (gcd) 10
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: December 19th 2008, 04:44 PM
  3. Divisibility (gcd) 9
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: December 19th 2008, 01:12 PM
  4. Divisibility (gcd) 8
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: December 19th 2008, 03:53 AM
  5. Divisibility
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: December 14th 2008, 09:24 AM

Search Tags


/mathhelpforum @mathhelpforum