Time Jumper

Number Theory Level 2

You've just got a magic watch, which can make you "jump" into 7 hours in the future with each tap on the watch.

Starting from 7 A.M., after tapping for some times, the watch's arms wind around wildly, ending with the short arm pointing at 6 though you're still too dizzy from time travel to know whether it's day or night.

What is the least possible number of taps on the watch?


The answer is 5.

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Worranat Pakornrat by Worranat Pakornrat , 36, Thailand

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1 solution

Let the least possible number of taps be n n . Since a clock uses arithmetic modulo 12. We can write:

7 n + 7 ≡ 6 (mod 12) 7 n ≡ − 1 (mod 12) 7 n ≡ 11 (mod 12) 7 ( 5 ) ≡ 11 (mod 12) ⟹ n = 5 \begin{aligned} 7n+7 & \equiv 6 \text{ (mod 12)} \\ 7n & \equiv -1 \text{ (mod 12)} \\ 7n & \equiv 11 \text{ (mod 12)} \\ \ 7(5) & \equiv 11 \text{ (mod 12)} \\ \implies n & = \boxed{5} \end{aligned}

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