14^1 mod 15 = 14
14^2 mod 15 = 1
14^3 mod 15 = 14
14^4 mod 15 = 1
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We can see that 14^(2n+1) mod 15 = 14 and 14^(2n) mod 15 = 1
So that 14^15 mod 15 = 14. :D
Now take an attempt on this problem:
\(14^{(1+2+3+4+5+6+...+107)}\pmod {15}=??\)
14^15 mod 15
\(\begin{array}{|rcll|} \hline \mathbf{ 14^{15} \pmod {15} =\ ? } \\\\ && 14^{15} \pmod {15} \quad &| \quad 14 \equiv -1 \pmod {15} \\ &\equiv& (-1)^{15} \pmod {15} \quad &| \quad (-1)^{15} = -1 \\ &\equiv& -1 \pmod {15} \\ &\equiv& -1 +15 \pmod {15} \\ &\equiv& 14 \pmod {15} \\\\ \mathbf{ 14^{15} \pmod {15} = 14 } \\ \hline \end{array} \)