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Math Help - Find the original number

  1. #1
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    Find the original number

    Dr Jenson writes a number on the whiteboard. He then writes 221 to the right hand side of this number, which ends in a new number which is a multiple of the old number (without the 221). What is the original number - find all possible solutions.

    I don't know if this fits under algebra, but thanks to anyone that helps.
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  2. #2
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    Hello, BG5965!

    Dr. Jenson writes a number on the whiteboard.
    He then writes 221 to the right hand side of this number,
    which ends in a new number which is a multiple of the old number.
    What is the original number? -- Find all possible solutions.

    Let N be the original number.

    If 221 is appended to the right end, the new number is: . 1000N + 221

    If this is a multiple of N, we have: . 1000N + 221 \:=\:k\!\cdot\!N . for some integer k.

    Then we have: . N \;=\;\frac{221}{k-1000}


    Since N is an integer, k-1000 must be a factor of 221.

    The only factors of 221 are: . 1, 13, 17, 221


    Therefore: . \begin{array}{ccccccc}<br />
k - 1000 \:=\: 1 & \Rightarrow & k \:=\: 1001 & \Rightarrow & \boxed{N \:=\:221} \\<br />
k-1000 \:=\: 13 & \Rightarrow & k \:=\: 1013 & \Rightarrow & \boxed{N \:=\:17}\\<br />
k-1000 \:=\: 17 & \Rightarrow & k \:=\: 1017 & \Rightarrow & \boxed{N \:=\:13}\\<br />
k-1000 \:=\: 221 & \Rightarrow & k \:=\: 1221 & \Rightarrow & \boxed{N \:=\:1} \end{array}

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