2006 UNCO Math Contest II Problems/Problem 3

Problem

The first 14 integers are written in order around a circle.

Starting with 1, every fifth integer is underlined. (That is $1,6,11,2,7,\ldots$ ). What is the $2006^{th}$ number underlined?

$[asy] draw(unitcircle,black); pair A; for(int j=1;j<15;++j){ A=dir(90-(j-1)*(360/14)); MP(string(j),A,A); } [/asy]$

Solution

The way the numbers cycle are $1, 6, 11, 2, 7, 12, 3, 8, 13, 4, 9, 14, 5, 10$ for a total of 14 numbers. $2006\div14=143R4$ , with a remainder of 4. The fourth number in the cycle is 2, so the answer is $\boxed2$

2006 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10
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2006 UNCO Math Contest II Problems/Problem 3

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