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Move the 2 - double the number (Posted on 2002-04-19) Difficulty: 3 of 5
A certain number ends with the digit 2. Moving the 2 from the end of the number to its front doubles it. Can you find this number?

(Hint: it's quite large)

See The Solution Submitted by levik    
Rating: 4.1111 (9 votes)

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Solution Another method | Comment 5 of 10 |
Rather than long division, it is possible to solve by filling in a multiplication problem.

We know that the first digit must be a 1, since it is half of a number starting with 2. So, a multiplication problem, with ... representing the missing digits in the product and in the multiplicand, would be:

1...2
*  2
21...

Therefore, the last digit in the product must be a 4 (from 2*2):

1...42
*    2
    4
21...4

The next to last digit is an 8 (4*2):

1...842
*    2
      4
    8
21...84

And then a 6 (from 8*2=16):

1...6842
*      2
      4
      8
    16 
21...684

Continuing thus:

1...36842
*      2
        4
      8
    16
    12 
21...3684

1...736842
*        2
        4
        8
      16
    12
    6 
21...73684

1...4736842
*        2
          4
        8
      16
      12
      6
    14   
21...4736842

1...94736842
*          2
          4
          8
        16
      12
      6
    14
    8     
21...9473684

1...894736842
*          2
            4
          8
        16
        12
        6
      14
      8
    18     
21...89473684

1...7894736842
*            2
            4
            8
          16
        12
        6
      14
      8
    18
    16       
21...789473684

1...57894736842
*            2
              4
            8
          16
          12
          6
        14
        8
      18
    16
    14       
21...5789473684

The next digit we find will be a 1 (the zero from 5*2=10 plus the carried 1), which is what we are looking for. However, there is another 1 carried over from this, so that will not do. We will keep going until we find a 1 with no carry over.

1...157894736842
*              2
              4
              8
            16
          12
          6
        14
        8
      18
      16
    14
    10         
21...15789473684

1...3157894736842
*              2
                4
              8
            16
            12
            6
          14
          8
        18
      16
      14
    10
    2         
21...315789473684

1...63157894736842
*                2
                4
                8
              16
            12
            6
          14
          8
        18
        16
      14
      10
      2
    6           
21...6315789473684

1...263157894736842
*                2
                  4
                8
              16
              12
              6
            14
            8
          18
        16
        14
      10
      2
      6
    12           
21...26315789473684

1...5263157894736842
*                  2
                  4
                  8
                16
              12
              6
            14
            8
          18
          16
        14
        10
        2
      6
    12
    4             
21...526315789473684

1...05263157894736842
*                  2
                    4
                  8
                16
                12
                6
              14
              8
            18
          16
          14
        10
        2
        6
      12
      4
    10             
21...0526315789473684

The next digit in the number will be a 1 (from 0*1=0 added to the carried 1). Since there is no carry over from the doubled zero, we can end here. The final multiplication is:

105263157894736842
*                2
                4
                8
              16
            12
            6
          14
          8
        18
        16
      14
      10
      2
    6
  12
  4
10
2               
210526315789473684

  Posted by DJ on 2003-06-25 16:07:43
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