how to calculate 14^15 mod 15

14^15 mod 15  = 14. Use the calculator here. First calculate: 14^15 and then press "mod" key followed by 15. You should get 14 as the remainder.

14¹⁵+1=(14+1)(14¹⁴-14¹³+14¹²-......-14+1)=15*(14¹⁴-14¹³+14¹²-......-14+1)= 14¹⁵+1 is divisible by 15

therefore, 14¹⁵ mod 15=15-1=14

Another way to solve it is this one:

${14}^{15}={(15-1)}^{15}={15}^{15}*\begin{pmatrix} 15\\ 0 \end{pmatrix} -{15}^{14}*\begin{pmatrix} 15\\ 1 \end{pmatrix} +.....+15*\begin{pmatrix} 15\\ 14 \end{pmatrix}- 1*\begin{pmatrix} 15\\ 15 \end{pmatrix}=15*({15}^{14}*\begin{pmatrix} 15\\ 0 \end{pmatrix}- {15}^{13}*\begin{pmatrix} 15\\ 1 \end{pmatrix}+....+1*\begin{pmatrix} 15\\ 14 \end{pmatrix})-1$

Here is another way

14^1=14=    -1mod15

14^2=196= +1 mod15

14^3=2744 = mod(2744,15) = 14 = -1

114^4 = 38416 = mod(38416,15) = 1

We have a pattern

$For\;\;n\in \text{Positive integers}\\ 14^{2n}=+1\\ 14^{2n-1}=-1 \\ so\\ 14^{15}\;\;mod\;15=-1\;\;or\;\;14$

https://en.wikipedia.org/wiki/Modular_exponentiation

how to calculate 14^15 mod 15

see link: https://web2.0calc.com/questions/i-have-the-same-problem-like-the-guest#r4