Find the remainder when $3^{101}$ is divided by $10$
Ans: $3$
My approach 1: Remainder theorem
When i apply remainder theorem I solve like $3$^$2$ . $50$+$1$/$3^2$-$(-1)$=$3$($-1)^{50}$=-$3$/$10$=$3$ @Edit
Approach2: Pattern method:
$3^4$ . $25$ + $1$=$3^\frac{1}{10}$=$3$
I am getting different answers through both approaches.
What is my mistake in †he first method and which to prefer while solving problems?
Best Answer
Hint : $3^2 = -1 $ (mod $10$).Hence what can you say about $3^{100}$ ?
Edit: By above observation ,$3^{100}=(3^2)^{50} =1$ (mod10).Hence what can you say about $3^{101}$ ?