A smart way to do this question.

Question

Let $S=\{0,1,2,\dotsc,25\}$

And $T=\{n\in S : n^2+3n+2\text{ is divisible by }6\}$

Then the number of elements in $T$ is?

One way I know is to factorise it as $(n+1)(n+2)$.

And then put each $n$ and check whether we get a $2$ and $3$ or their multiples in the factors.

However it is a bit time consuming.

So I ask for a smarter(quicker) way if any, to this?

Answer 1

As $(n+1)(n+2)$ is product of two consecutive integers, $2\mid(n+1)(n+2)$

Again, $3\mid(n+1)(n+2)\implies$

either $3\mid(n+1)\iff n\equiv-1\equiv2\pmod3$

or $3\mid(n+2)\iff n\equiv-2\equiv1\pmod3$

Also, though not required here, $3$ can not divide both as $(n+2,n+1)=1$