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Move the 6, Quadruple the Number (Posted on 2003-09-04) Difficulty: 4 of 5
In Move the 2, Double the Number, we found a number that ended in two, for which moving the two to the beginning of the number doubled its value.

For this problem, we have a number with 6 as the last (right-most) digit.
If we erase the 6 and put it on the left end of the number (for example, 936 would become 693), then we have a number four times our original number (we see that 936 doesn't work, of course).

What is the smallest number that fits this condition?

What is the second smallest number that works?

What is the tenth smallest number that works?

  Submitted by DJ    
Rating: 4.1667 (12 votes)
Solution: (Hide)
153846

First, the first digit must be a 1. Any number starting with 6, when divided by 4, will yield a number starting with 1 (with the same number of digits as the original number).

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...6
×   4
61...
So, the last digit of the product must be 4 (the last digit of 6×4):
1...46
×    4
    24
61...4
So the next to last digit of the solution must be 8 (6×4 yields 24, with a 4 in the ones place and a two carried over, to be added to the 8 from 4×4=16):
1...846
×     4
     24
    16 
61...84
So the third to last digit must be 3 (the carried 1 from 16, plus the last digit of 8×4=32):
1...3846
      ×4
      24
     16
    32  
61...384
So the next digit must be a 5 (3 from 32, added to the 2 from 3×4=12):
1...53846
×       4
       24
      16
     32
    12   
61...5384
The next digit must be a 1 (carried over from 12, and 5×4=20, so there is nothing to add to it), which is what we were looking for:
153846
×    4
    24
   16
  32
 12
20
4    
615384
Therefore, 153846 is the lowest number that works.

Moving Right Along:
Actually, we could keep going with the last method:
1...6153846
×         4
         24
        16
       32
      12
     20
     4     
61...615384

1...46153846
×          4
          24
         16
        32
       12
      20
      4
    24      
61...4615384
We are back to the step we started with, and it is clear the pattern will repeat itself.
After any number of repetitions, we can let the 1 at the beginning be the 1 used in the product, so all solutions have the form: 153846, 153846153846, 153846153846153846, etc.

Therefore, the second lowest number that works is 153846153846.

Similarly, the tenth smallest number that works must be 153846153846153846153846153846153846153846153846153846153846.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
SolutionSolutionDej Mar2007-11-28 09:29:14
answerK Sengupta2007-08-31 23:26:03
SolutionSolutionNick Hobson2003-10-10 21:28:37
SolutionNot for SIlver Knight's EyesFatBoy2003-09-05 07:44:25
solutionEric2003-09-05 00:27:27
SolutionsolutionCharlie2003-09-04 14:47:24
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