A fifty digit positive integer satisfies all these given conditions:
- The number is constituted entirely by the digit 1 with the exception of the units digit.
- The number is divisible by 53.
Determine the units digit.
Extra Challenge: Solve this puzzle without using a computer-assisted method.
mod 53:
10^2 = 100 - 53 = 47
10^4 = 47^2 = 2209 = 36
10^5 = 360 = 42
10^10 = 42^2 = 1764 = 15
10^50 = 15^5 = 759375 = 44
10^50-1 = 43
But this is the modular value of 50 9's, not the 50 1's we need. If we knew how to take modular multiplicative inverses we'd be set. Google search finds a few methods, the most promising involving the same Euclidean algorithm for finding GCD, and factors that can be used in finding that, but even that looks like it needs a computer to be manageable rather than daunting.
One could construct a list of powers of 10 mod 53 going from 10^1 through 10^49 and add them up to see how much extra is needed. Daunting!
Easier to use the computer as a calculator:
>> s=sym(10)^50-1
s =
99999999999999999999999999999999999999999999999999
>> s=s/9
s =
11111111111111111111111111111111111111111111111111
>> mod((s-1),53)
ans =
45
So 49 1's and a zero has modular value 45.
We need 8 more to get to 53 which is equivalent to zero.
The last digit is 8.
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Posted by Charlie
on 2022-05-04 12:50:44 |
computer solution
