M ath O n D iscs

Number Theory Level 3

Steve places a counter at 0 on the diagram. On his first move, he moves the counter $1^1$ step clockwise to 1. On his second move, he moves $2^2$ steps clockwise to 5. On his third move, he moves $3^3$ steps clockwise to 2. He continues in this manner, moving $n^n$ steps clockwise on his $n^\text{th}$ move. At which position will the counter be after 1234 moves?

Source: CMO Cayley Contest (Grade 10).

The answer is 7.

2 solutions

Giorgos K.
May 13, 2018

$Mathematica$ code

Mod[Sum[n^n,{n,1234}],10]

returns $7$

Pigeon Pigeon
Mar 2, 2018

Using modular arithmetic, we can find how any input of n (which output is n^2) *is always broken down to a value in between 0-9. We must always add 1 to the modular value * because after the first move everything starts off from 1. nk mod 10 = k This formula tells us that any value mod 10 always equals the ones digit. Using that info, we simplify the problem down to finding the ones digit in 1234^1234. 1234^1234 mod 10 = ...4^1234 Using basic rules of powers of 4: 4^even = ...6 4^odd = ...4 We find that 1234^1234 ends in 6. Applying the simple rule of modular arithmetic, we find that 1234^1234 mod 10 = 6

*The formula for this problem however is (n^n mod 10)+1 Just add 1+6 and you get the answer, 7. *

M ath O n D iscs

The answer is 7.

2 solutions

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