Question

whats the last digit of 7 to the power of 200

Answer 1

Last digit in 7^200
Here 7 is base and 200 is index
to find unit digit, the general method is
Base / 4 find remainder, If remainder
is 1 then unit digit in the answer is (Base unit digit)^1
is 2 then unit digit in the answer is (Base unit digit)^2
is 3 then unit digit in the answer is (Base unit digit)^3
is 4 then unit digit in the answer is (Base unit digit)^4
Here,
200/4 remainder is zero so unit digit in the answer
= 7^4 = 7 × 7 × 7 × 7 = unit digit = 1

Answer 2

so the answer to that is 1?

Answer 3

by binomial theorm
\(\large\tt \begin{align} \color{black}{ rem\implies \dfrac{7^{200}}{10}\\~\\
rem\implies \dfrac{49^{100}}{10}\\~\\
rem\implies \dfrac{(50-1)^{100}}{10}\\~\\
rem\implies \dfrac{(5\times 10-1)^{100}}{10}\\~\\
rem\implies \dfrac{(-1)^{100}}{10}\\~\\
rem\implies \dfrac{1}{10}\\~\\
\Large rem\implies 1\\~\\
}\end{align}\)

Answer 4

nice :)

Answer 5

\(7^{200}\mod(10)=\\(7^2)^{100} \mod(10) =\\49^{100}\mod(10) =\\ 9^{100}\mod(10)=\\(9^2)^{50}\mod(10)=\\1^{50}\mod(10)=\\1\mod(10)\)