Find the remainder when 4⁹⁶ is divided by 6.
a) 0
b) 2
c) 3
d) 4

Answer: d
Explanation: $$\frac{{{4^{96}}}}{6},$$ we can write it in this form
$$\frac{{{{\left( {6 – 2} \right)}^{96}}}}{6}$$
Now, Remainder will depend only the powers of -2.
$$\frac{{{{\left( { – 2} \right)}^{96}}}}{6},$$   it is same as
$$\frac{{{{\left( {{{\left[ { – 2} \right]}^4}} \right)}^{24}}}}{6},$$   it is same as
$$\frac{{{{\left( {16} \right)}^{24}}}}{6}$$
$$\frac{{\left( {16 \times 16 \times 16 \times 16{\kern 1pt} ……{\kern 1pt} 24{\kern 1pt} {\text{times}}} \right)}}{6}$$
On dividing individually 16 we always get a remainder 4.
$$\frac{{\left( {4 \times 4 \times 4 \times 4{\kern 1pt} ……{\kern 1pt} 24{\kern 1pt} {\text{times}}} \right)}}{6}$$
Required Remainder = 4
Note: When 4 has even number of powers, it will always give remainder 4 on dividing by 6.