Question

Find the remainder when ${10}^{100}$ is divided by 7

• A. 4
• B. 3
• C. 2
• D. –1
• E. none of these

Answer 1

The correct option is A 4

Option (a)
Euler's number of 7 ( a prime number) is $7-1=6$. make the numerator power the nearest
multiple of 6. The question changes to
$10\frac{96}{7}{\mid }_R\times 10\frac{4}{7}{\mid }_R$
part a will yield a remainder $1$ and $10\frac{4}{7}{\mid }_R$ can be found using frequency method
$10\frac{1}{7}{\mid }_R=3$
$10\frac{2}{7}{\mid }_R=2$
$10\frac{3}{7}{\mid }_R=\frac{6}{7}{\mid }_R=-1$
$10\frac{4}{7}{\mid }_R=\frac{4}{7}{\mid }_R=4$
answer = option a