Question

What are the two rightmost digits of 3^33+ 33^333?

Answer 1

The last two right most digit is 36

Step- by step -Explanation:

$3^{33} =81^{8}.3$

$=(100-19)^{8} .3$

$=(-19)^{8}.3$

$=(-19)^{5} .(-19)^{3} .3$

$=(1-20)^{5} .(1-20)^{3} .3$

$=1.(1-3.20).3$

$=(-59).3$

$=-177$

$=23(mod100).$

$33^{333} =(30+3)^{333}$

$=3^{333} +333.3^{332} .30$

$=3^{333} .(1+3330)$

$=3^{300} .3^{33} .31$

$=(3^{20} )^{15}.3^{33} .31$

$=1.23.31$

$=600+20+90+3$

=13

Hence

$3^{33} +33^{33} =23+13=36(mod100) .$

Therefore, the last two digits are 36 .