Number Puzzle Based on Multiplication Ideas

Date: 08/07/2005 at 11:29:58
From: Kate
Subject: Find the smallest natural number n......

Find the smallest natural number n which has the following properties:

* Its decimal representation has 6 as the last digit
* If the last digit 6 is erased and placed in front of the remaining 
digits, the resulting number is 4 times as large as the original 
number n.

I know that the natural number n has ...46 as last 2 digits.  When the 
digit 6 is shifted to front, then it will be 6...4.  Also, before 
shifting the digit 6, the most front digit must be less than 6. 

That's what I have realized and tested, but I still can't work it out.

Date: 08/08/2005 at 00:15:29
From: Doctor Greenie
Subject: Re: Find the smallest natural number n......

Hi, Kate --

You are off to a great start when you noted that the last 2 digits of
the number must be "46".

How did you determine that?  Presumably, you thought something like 
this:

  ???????6
     x   4
  --------
  ???????4

Then, according to the statement of the problem, the next-to-last 
digit in the number must be "4".  So now you had

  ??????46
    x    4
  --------
  ???????4

But we can continue using this same reasoning over and over until we 
find a solution to the problem.  Because we know the last two digits 
of the original number are "46", we can perform the multiplication 
by 4 of these two final digits to determine that the second digit from 
the right in the product is "8":

  ??????46
    x    4
  --------
  ??????84

Now we know the last two digits of the product; and according to the 
rules of the problem, the "8" is the third digit from the right in the 
original number.  Then, by performing the multiplication by 4 of the 
digits we now know of the original number, the third digit from the 
right in the product is uniquely determined:

  ?????846
    x    4
  --------
  ?????384

And we just continue this process--multiplying the last digit we found 
in the original number by 4 to find the next digit to the left in the 
product, and copying that digit as the next digit to the left in the 
original number:

  ????3846
    x    4
  --------
  ????5384

  ???53846
    x    4
  --------
  ???15384

  ??153846
    x    4
  --------
  ??615384

We now have a "6" as the first digit of the product; this means we are 
done.  The original number is 153846; multiplied by 4 the product is 
615384.

We can also work the problem in the opposite direction by a similar 
process.  You noted in your original message that the first digit of 
the original number must be "less than 6".  In fact, if the original 
number multiplied by 4 is to have "6" as the first digit, then the 
first digit of the original number must be "1".

So we have

  1????????
     x    4
  ---------
  6????????

But now, according to the rules for the problem, the "1" which is the 
first digit of the original number must be the second digit of the 
product:

  1????????
     x    4
  ---------
  61???????

Now, by dividing the portion of the product we know by 4, we can see

  61/4 = 15 (plus a remainder)

so the second digit of the original number must be "5"--and that means 
the third digit of the product is "5":

  15???????
     x    4
  ---------
  615??????

Again, we can repeat this process until we find the solution to the 
problem:

  615/4 = 153 (plus a remainder) so next digit is "3":

  153??????
     x    4
  ---------
  6153?????

  6153/4 = 1538 (plus a remainder) so next digit is "8":

  1538?????
     x    4
  ---------
  61538????

  61538/4 = 15384 (plus a remainder) so next digit is "4":

  15384????
     x    4
  ---------
  615384???

  615384/4 = 153846 (with no remainder)

Since there is no remainder, we are done.  Again we have (of course) 
found the same answer: the original number is 153846, and the product 
when multiplied by 4 is 615384.

Please write back if you have questions on any of this.

- Doctor Greenie, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 08/08/2005 at 06:05:07
From: Kate
Subject: Thank you (Find the smallest natural number n......)

Doctor Greenie,

Thanks for your help in solving this question.  I appreciate it,
really!  This is really a good place to learn math!  I love it!