2007 iTest Problems/Problem 56

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The following problem is from the Ultimate Question of the 2007 iTest, where solving this problem required the answer of a previous problem. When the problem is rewritten, the T-value is substituted.

Problem

In the binary expansion of $\dfrac{2^{2007}-1}{2^{225}-1}$, how many of the first $10,000$ digits to the right of the radix point are $0$'s?

Solution

We can approach this problem by using long division in base 2 because long division takes advantage of regrouping. The number $2^{2007} - 1$ has $2007$ ones while the number $2^{225} - 1$ has $225$ ones.

                        
   [225 ones])[2007 ones]

Note that $2^{2007} - 1$ can be rewritten as $(2^{225} - 1)(2^{1782} + 2^{1557} + 2^{1332} + \cdots 2^{207}) + 2^{207} - 1,$ so the remainder when $2^{2007} - 1$ is divided by $2^{225} - 1$ is the same when $2^{207} - 1$ is divided by $2^{225} - 1.$

                      
   [225 1’s])[207 1’s]

There are not enough digits for $2^{207} - 1$ to divide evenly into $2^{225} - 1,$ so we need to bring down more zeroes. Since $2^{225} - 2^{18} < 2^{225} - 1 < 2^{226} - 2^{19}$, we need to bring down 19 zeroes, resulting in 18 zeroes to the right of the radix point.

                    0.[18 0’s]
   [225 1’s])[207 1’s][18 0’s]0

Now we can subtract $2^{225} - 1$ in the long division.

                    0.[18 0’s]1
   [225 1’s])[207 1’s][18 0’s]0
            -[206 1’s][18 1’s]1
             [206 1’s][18 0’s]1

We can bring down more zeroes and repeat the iteration.

                    0.[18 0’s]111
   [225 1’s])[207 1’s][18 0’s]0
            -[206 1’s][18 1’s]1
             [206 1’s][18 0’s]10
            -[205 1’s][18 1’s]11
             [205 1’s][18 0’s]110
            -[204 1’s][18 1’s]111
             [204 1’s][18 0’s]111

Notice that no zeroes are placed to the right of the last values after each iteration. Also, there is a pattern after doing the subtraction in each iteration, where there are $n$ ones followed by $18$ zeroes and $207-n$ ones.


To confirm this, we note that in each iteration, we multiply by $2$ and subtract $2^{2007} - 1.$ Doing this results in

  [n   1’s][18 0’s][207-n 1’s]0
 -[n-1 1’s][18 1’s][207-n 1’s]1
  [n-1 1’s][18 0’s][207-(n-1) 1’s]

Thus, we see that the pattern holds. In fact, tthe pattern repeats $206$ times, and after that, the number from the subtraction just has $207$ consecutive ones. That means the digits repeats every $225$ digits, and $18$ of these digits are zeroes.


That means of the $9900$ digits past the radix point, $792$ of them are zeroes. After $9918$ digits, there are $810$ zeroes, and since the $82$ digits after are ones, we confirm that there are $\boxed{810}$ zeroes of the first $10,000$ digits past the radix point.

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

See Also

2007 iTest (Problems)
Preceded by:
Problem 55
Followed by:
Problem 57
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