When $(5k+1)^2$ is divided by $5$,the remainder is . . . . . . .

Prove that $3 + 2\sqrt{5}$ is irrational.

Find the Least Common Multiple $(LCM)$ for the following pairs and match them with the correct options:

$Q.1.$ $\text{LCM}(7, 49)$	$A. 50$
$Q.2.$ $\text{LCM}(10, 25)$	$B. 49$
	$C. 51$
	$D. 50$

The last digit of $2^{m} \cdot 5^{n}$ (where $m, n \in N$) is . . . . . . .

For the decimal expansion of a rational number $\frac{p}{q}$ to be terminating,if $q = 2^m \cdot 5^n$,then $m, n \in$ . . . . . . ($R$,$N \cup \{0\}$,$Z$)

Prove that $7\sqrt{5}$ is an irrational number.