According to Euclid's division lemma,for $a$ and $5$,if $a = 5q + r$ is unique,then $r = \ldots$ is not possible.

Find the $g.c.d.$ (Greatest Common Divisor) of $75$ and $243$ using Euclid's division algorithm.

For any positive integer $n,$ prove that $n^{3}-n$ is divisible by $6.$

The $\text{g.c.d.}$ (greatest common divisor) of any two consecutive natural numbers is:

State whether the following rational number has a terminating decimal expansion or not. If it has a terminating decimal expansion,find it: $\frac{17}{8}$

Using Euclid's division algorithm,find the largest number that divides $1251, 9377$ and $15628$ leaving remainders $1, 2$ and $3$,respectively.