Question

Find the remainder when ${10}^{1000}$ is divided by 7.
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Answer 1

We can apply Euler's Remainder Theorem here as 10 and 7 are co-primes.
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 - ${10}^{996}\times {10}^4$ divided by 7.
${10}^{996}$ = ${10}^{166\times 6}$ will leave a remainder of 1 when divided by 7.
${10}^4$ when divided by 7 leaves a remainder of 4.
So, actual remainder will be $1\times 4$ = 4.