17. Find the value of $R$, given that the numbers $Q$ and $R$ are defined as:
$$
Q=202^{1}+20^{21}+2^{021} \text {; }
$$
$R$ is the remainder when $Q$ is divided by 1000 .
SolUtion
354

The term $20^{21}$ is divisible by $10^{21}$ and therefore is divisible by 1000 , meaning it will not contribute to the remainder on division by 1000 .
The term $2^{021}=2^{21}=2^{10} \times 2^{10} \times 2=1024 \times 1024 \times 2=(1000+24) \times(1000+24) \times 2$.
When multiplied out, all terms involving a 1000 can be ignored.
So this contributes $24 \times 24 \times 2=1152$, or equivalently 152 , to the remainder.
Hence the total remainder is $R=202^{1}+152=354$.