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Move the 3, Triple the number (Posted on 2004-03-01) Difficulty: 2 of 5
If you moved the 3 at the end of a certain large number to the beginning, it would be the same as multiplying it by 3. What is the number?

See The Solution Submitted by Gamer    
Rating: 3.2000 (5 votes)

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Solution Longhand... | Comment 5 of 17 |
First, the first digit must be 1. Any number starting with 3, divided by 3, will yield a number that starts with a 1.

Therefore, a good place to start would be to write out the multiplication, using ... to indicate the unknown digits in both the product and the multiplicand.

1...3
<U>   3</U>
31...
So, the last digit of the product must be 9 (33):
1...93
<U>    3</U>
<U>     9</U>
31...9
The next to last digit must be 7 (39 is 27):
1...793
<U>     3</U>
      9
<U>    27 </U>
31...79
The third-to-last digit is 3 (the 2 carried over from 27, plus the last digit of 73=21):
1...3793
<U>      3</U>
       9
     27
<U>    21  </U>
31...379
Continuing thus:
1...13793
<U>       3</U>
        9
      27
     21
<U>    9   </U>
31...1379

1...413793
<U>       3</U>
        9
      27
     21
     9
<U>    3   </U>
31...41379

1...413793
<U>       3</U>
        9
      27
     21
     9
    3   </U>
31...41379

1...413793
<U>       3</U>
        9
      27
     21
     9
    3   </U>
31...41379

1...2413793
<U>         3</U>
          9
        27
       21
       9
      3
<U>    12     </U>
31...241379

1...72413793
<U>          3</U>
           9
         27
        21
        9
       3
     12
<U>     6      </U>
31...7241379

1...172413793
<U>           3</U>
            9
          27
         21
         9
        3
      12
      6
<U>    21       </U>
31...17241379

1...5172413793
<U>            3</U>
             9
           27
          21
          9
         3
       12
       6
     21
<U>     3        </U>
31...517241379

1...55172413793
<U>             3</U>
              9
            27
           21
           9
          3
        12
        6
      21
      3
<U>    15         </U>
31...5517241379

1...655172413793
<U>              3</U>
               9
             27
            21
            9
           3
         12
         6
       21
       3
     15
<U>    15          </U>
31...65517241379

1...9655172413793
<U>               3</U>
                9
              27
             21
             9
            3
          12
          6
        21
        3
      15
     15
<U>    18           </U>
31...965517241379

1...89655172413793
<U>                3</U>
                 9
               27
              21
              9
             3
           12
           6
         21
         3
       15
      15
     18
<U>    27            </U>
31...8965517241379

1...689655172413793
<U>                 3</U>
                  9
                27
               21
               9
              3
            12
            6
          21
          3
        15
       15
      18
     27
<U>    24             </U>
31...68965517241379

1...0689655172413793
<U>                  3</U>
                   9
                 27
                21
                9
               3
             12
             6
           21
           3
         15
        15
       18
      27
     24
<U>    18              </U>
31...068965517241379

1...20689655172413793
<U>                   3</U>
                    9
                  27
                 21
                 9
                3
              12
              6
            21
            3
          15
         15
        18
       27
      24
     18
<U>     0               </U>
31...2068965517241379

1...620689655172413793
<U>                    3</U>
                     9
                   27
                  21
                  9
                 3
               12
               6
             21
             3
           15
          15
         18
        27
       24
      18
      0
<U>     6                </U>
31...62068965517241379

1...8620689655172413793
<U>                     3</U>
                      9
                    27
                   21
                   9
                  3
                12
                6
              21
              3
            15
           15
          18
         27
        24
       18
       0
      6
<U>    18                 </U>
31...862068965517241379

1...58620689655172413793
<U>                      3</U>
                       9
                     27
                    21
                    9
                   3
                 12
                 6
               21
               3
             15
            15
           18
          27
         24
        18
        0
       6
     18
<U>    24                  </U>
31...5862068965517241379

1...758620689655172413793
<U>                       3</U>
                        9
                      27
                     21
                     9
                    3
                  12
                  6
                21
                3
              15
             15
            18
           27
          24
         18
         0
        6
      18
     24
<U>    15                   </U>
31...75862068965517241379

1...2758620689655172413793
<U>                        3</U>
                         9
                       27
                      21
                      9
                     3
                   12
                   6
                 21
                 3
               15
              15
             18
            27
           24
          18
          0
         6
       18
      24
     15
<U>    21                    </U>
31...275862068965517241379

1...82758620689655172413793
<U>                         3</U>
                          9
                        27
                       21
                       9
                      3
                    12
                    6
                  21
                  3
                15
               15
              18
             27
            24
           18
           0
          6
        18
       24
      15
     21
<U>     6                     </U>
31...8275862068965517241379

1...482758620689655172413793
<U>                          3</U>
                           9
                         27
                        21
                        9
                       3
                     12
                     6
                   21
                   3
                 15
                15
               18
              27
             24
            18
            0
           6
         18
        24
       15
      21
      6
<U>    24                      </U>
31...48275862068965517241379

1...4482758620689655172413793
<U>                           3</U>
                            9
                          27
                         21
                         9
                        3
                      12
                      6
                    21
                    3
                  15
                 15
                18
               27
              24
             18
             0
            6
          18
         24
        15
       21
       6
     24
<U>    12                       </U>
31...448275862068965517241379

1...34482758620689655172413793
<U>                            3</U>
                             9
                           27
                          21
                          9
                         3
                       12
                       6
                     21
                     3
                   15
                  15
                 18
                27
               24
              18
              0
             6
           18
          24
         15
        21
        6
      24
     12
<U>    12                        </U>
31...3448275862068965517241379

1...034482758620689655172413793
<U>                             3</U>
                              9
                            27
                           21
                           9
                          3
                        12
                        6
                      21
                      3
                    15
                   15
                  18
                 27
                24
               18
               0
              6
            18
           24
          15
         21
         6
       24
      12
     12
<U>     9                         </U>
31...03448275862068965517241379

1...1034482758620689655172413793
<U>                              3</U>
                               9
                             27
                            21
                            9
                           3
                         12
                         6
                       21
                       3
                     15
                    15
                   18
                  27
                 24
                18
                0
               6
             18
            24
           15
          21
          6
        24
       12
      12
      9
<U>     0                          </U>
31...103448275862068965517241379
Finally, we have a digit 1 without a carry (the other ones were actually 11). This 1 can be used as the first digit, giving the final answer:
1034482758620689655172413793
<U>                          3</U>
                           9
                         27
                        21
                        9
                       3
                     12
                     6
                   21
                   3
                 15
                15
               18
              27
             24
            18
            0
           6
         18
        24
       15
      21
      6
    24
   12
  12
  9
 0
<U>3                           </U>
3103448275862068965517241379

Of course, that is just the smallest answer. If we didn't make that 1 be the first digit, and kept going, the whole process would just repeat itself. The next smallest number, then, would be 10344827586206896551724137931034482758620689655172413793, and so on.
  Posted by DJ on 2004-03-01 11:52:19
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