Question

For integer n > 1, the digit at units place in the number ${\sum }_{r=0}^{100}\angle r+2^{4n}$ is

• A. 1
• B. 2
• C. 4

Answer 1

For n > 1, $2^n$ is a multiple of 4
Let $2^n=4K,KϵN$
The digit at units place is $2^4$ is 6
$\Rightarrow$ The digit at units place is $2^{4K}$ i.e. $2^{4n}$ is 6
The digit at units place in each of $\angle 5,\angle 6,\dots \dots \angle 100$ is 0, but $\angle 0+\angle 1+\dots \dots +\angle 4=1+1+2+6+24=34$
$\Rightarrow$ The digit at units place in ${\sum }_{r=0}^{100}\angle r=4$
$\therefore$ The digit at units place in ${\sum }_{r=0}^{100}\angle r+2^{4n}=0$
$(\because 6+4=10)$