Question

What is the remainder left when $5^{589}$ is divided by 1000?

Answer 1

Pattern,

$5^1=55^2=255^3=1255^4=6255^5=3125$ (exponent is in the form of 4k+1)
$5^6=15625$
589 is in the form of 4k+1.
Remainder of any number more than 1000 will be the last three digits of the number.
Example: 1234 when divided by 1000 gives the remainder as 234.
Thus studying the pattern,
In, $5^{589}$ last three digits will be 125.
Thus the remainder will be 125.