Problem 26 · 2021 Math Kangaroo
Stretch
Number Theory
mod-10casework
My little brother has a 4-digit bike lock with the digits 0 to 9 on each part of the lock as shown. He started on the correct combination and turned each part the same amount in the same direction and now the lock shows the combination 6348. Which of the following CANNOT be the correct combination of my brother's lock?
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Answer: C
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Hint 1 of 2
Turning every wheel by the same amount in the same direction shifts each digit by the same value (mod 10).
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Hint 2 of 2
Subtract 6348 digit-by-digit (mod 10); a real start must give the same shift on all four digits.
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Approach: check for a constant per-digit shift mod 10
- Since each wheel turned the same amount, the true combination differs from 6348 by the same shift in every digit (mod 10).
- For 8560 the shifts are 2,2,2,2 and for the others they are equal too — except 4906, whose shifts 8,6,2,8 are not all the same.
- So 4906 cannot be the correct combination: choice C.
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