 # Find the smallest positive integer k satisfying the stated condition or explain no such k exists: 2^k -= 1 mod 14. BenoguigoliB 2021-01-04 Answered
Find the smallest positive integer k satisfying the stated condition or explain no such k exists:
${2}^{k}\equiv 1\text{mod}14$.
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Find ${2}^{k}\left(\text{mod}14\right)$ for some values of k.
${2}^{1}\equiv 2\left(\text{mod}14\right)$
${2}^{2}\equiv 4\left(\text{mod}14\right)$
${2}^{3}\equiv 8\left(\text{mod}14\right)$
${2}^{4}\equiv 2\left(\text{mod}14\right)$
${2}^{5}\equiv 4\left(\text{mod}14\right)$
......
So, ${2}^{k}\left(\text{mod}14\right)$ gives the value 2,4 or 8 for any positive integer k.
Hence, there exists no positive integer k such that ${2}^{k}\equiv 1\left(\text{mod}14\right)$.