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 Move the 3, Triple the number (Posted on 2004-03-01)
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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 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 (3×3):
```1...93
<U>×    3</U>
<U>     9</U>
31...9
```
The next to last digit must be 7 (3×9 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 7×3=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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