Question : A number $n$ when divided by 6, leaves a remainder of 3. What will be the remainder when $\left(n^2+5 n+8\right)$ is divided by 6?
Option 1: 1
Option 2: 3
Option 3: 5
Option 4: 2
Correct Answer: 2
Solution :
Euclid's Division Lemma,
Dividend = Divisor × Quotient + Remainder
Number = $n$
So, according to Euclid's Division Lemma
$n= 6q+ 3$ where $n$ is dividend and $q$ is quotient.
Also, $(n^2 + 5n + 8)$ when divided by 6 leaves a remainder, say $r$.
$(n^2 + 5n + 8)$
= $(6q+3)^3 + 5(6q+3) +8$
= $36q^2+ 9+ 36q+ 30q+ 15+ 8$
= $36q^2+ 66q+ 32$
Now, consider each of the terms above separately to see what the remainder is when they are divided by 6.
$36q^2$ is divisible by 6 as 6 is divisible by 6. Hence, the remainder is 0.
$66q$ is divisible by 6 as 66 is division by 6. Hence, the remainder is 0.
32 when divided by 6 leaves a remainder of 2.
$\therefore$ Remainder for $(n^2 + 5n + 8)$ is 2.
Hence, the correct answer is 2.
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