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 Move the 2 - double the number (Posted on 2002-04-19)
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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 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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