Find the last two digits in the decimal representation (base 10) of 17^20

Question

My attempt is in this image. But I want the answer to this question by using modulo congruence method since the method used by me involves tedious calculations.

Answer 1

Working modulo $100$,

$17^{20} \equiv 289^{10} \equiv (-11)^{10} \equiv (10+1)^{10} \overset{binom}{\equiv} \binom {10}1 10^1\cdot 1^9 + 1 \equiv 1 \pmod{100}$

So the last two digits are $01$.

Answer 2

Euler $\phi(5^2)=5\cdot4=20$ so $\,17^{20}\!\equiv 1\pmod{5^2}.\,$ Also $\,17^{20}\equiv 1^{20}\equiv 1\pmod 4,\ $ therefore $\,4,25\mid 17^{20}\!-1\,\Rightarrow\, 4\cdot 25\mid 17^{20}-1$

Answer 3

Oh, just to be different.

$17^{5} \mod 100 \equiv$

$(20 - 3)^5 \mod 100 \equiv$

$\sum_{k=0}^5 {5 \choose k} 20^k*(-3)^{5-k} \mod 100 \equiv$ [all but the last term is divisible by $20*5 = 100$]

$(-3)^{5} \mod 100$

So

$17^{20} \mod 100 \equiv$

$(-3)^{20} \mod 100 \equiv$

$9^{10} \mod 100 \equiv$

$(10 - 1)^{10} \mod 100 \equiv$

$\sum_{k=0}^{10}{10 \choose k}10^k(-1)^{10 - k} \mod 100 \equiv$ [all but the last term is divisible by $10*10 = 100$]

$\equiv 1 \mod 100$.

So last two digits are $01$.

....

So... tedious calculations need not be that tedious.

Answer 4

The units digit of $17^1$ is equal to $7$.
The units digit of $17^2$ is equal to $9$.
The units digit of $17^3$ is equal to $3$.
The units digit of $17^4$ is equal to $1$.

Calculating,

$\quad 17^4 = (17^2)(17^2) = (289)(289) \equiv (-11)(-11) \equiv 21 \pmod{100}$

Continuing,

$\quad 21^2 \equiv 41 \pmod{100}$
$\quad 21^3 \equiv 61 \pmod{100}$
$\quad 21^4 \equiv 81 \pmod{100}$
$\quad 21^5 \equiv \;\,1 \pmod{100}$

So,

$\quad 17^{20} = (17^4)^5 \equiv 21^5 \equiv 1 \pmod{100}$