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Find the remainder when $19^{729}$ when divided by 107?

My approach

$\frac{19^{723}}{107}$=$\frac{19^{636}}{107}$*$\frac{19^{87}}{107}$

by Euler theorem, we know that remainder of $\frac{19^{636}}{107}$ =1

so the whole problem turned into the following

$\frac{19^{87}}{107}$

now

19 divided by 107 leaves remainder 19

19*19 divided by 107 leaves remainder 40

19*19*19 divided by 107 leaves remainder 11

19*19*19*19 divided by 107 leaves remainder 102

so how to approach this question??

  • 0
    Is it $729$ or $723$??? I've answered for $729$. – 2017-02-16
  • 0
    if it is 729 then tell me your answer @barakmanos – 2017-02-16
  • 0
    It's right below your question! – 2017-02-16

1 Answers 1

-1

Note that:

$19^{1}\equiv\color\red{19}\pmod{107}\implies$

$19^{2}\equiv\color\red{19}^2\equiv361\equiv\color\red{40}\pmod{107}\implies$

$19^{4}\equiv\color\red{40}^2\equiv1600\equiv\color\red{102}\pmod{107}\implies$

$19^{8}\equiv\color\red{102}^2\equiv10404\equiv\color\red{25}\pmod{107}\implies$

$19^{16}\equiv\color\red{25}^2\equiv625\equiv\color\red{90}\pmod{107}\implies$

$19^{32}\equiv\color\red{90}^2\equiv8100\equiv\color\red{75}\pmod{107}\implies$

$19^{64}\equiv\color\red{75}^2\equiv5625\equiv\color\red{61}\pmod{107}\implies$

$19^{128}\equiv\color\red{61}^2\equiv3721\equiv\color\red{83}\pmod{107}\implies$

$19^{256}\equiv\color\red{83}^2\equiv6889\equiv\color\red{41}\pmod{107}\implies$

$19^{512}\equiv\color\red{41}^2\equiv1681\equiv\color\red{76}\pmod{107}$

Therefore:

$19^{729}\equiv$

$19^{512+128+64+16+8+1}\equiv$

$19^{512}\cdot19^{128}\cdot19^{64}\cdot19^{16}\cdot19^{8}\cdot19^{1}\equiv$

$76\cdot83\cdot61\cdot90\cdot25\cdot19\equiv$

$16449687000\equiv$

$56\pmod{107}$

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    If you want $19^{723}$, then just replace $19^{8}$ with $19^{2}$ (i.e., $25$ with $40$). – 2017-02-16