If-3-2019-k-is-a-multiple-of-11-find-the-smallest-positive-integer-k-

Question Number 124931 by ZiYangLee last updated on 07/Dec/20

If 3^{2019} + k is a multiple of 11,

find the smallest positive integer k .

Answered by MJS_new last updated on 07/Dec/20

3^{0} = 11 n + 1

3^{1} = 11 n + 3

3^{2} = 11 n + 9

3^{3} = 11 n + 5

3^{4} = 11 n + 4

3^{5} = 11 n + 1

3^{6} = 11 n + 3

\dots

\Rightarrow

3^{5 m} = 11 n + 1

3^{5 n + 1} = 11 n + 3

3^{5 m + 2} = 12 n + 9

3^{5 m + 3} = 11 n + 5

3^{5 m + 4} = 11 n + 4

2019 = 5 m + 4 \Rightarrow 3^{2019} = 11 n + 4 \Rightarrow k = 7

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