Let N = 1421 × 1423 × 1425. What is the remainder when N is divided by 12?

Remainder,
$$\eqalign{ & \frac{{1421 \times 1423 \times 1425}}{{12}} = R \cr & R \Rightarrow \frac{{5 \times 7 \times 9}}{{12}} \cr} $$

[Here, we have taken individual remainder such as 1421 divided by 12 gives remainder 5, 1423 and 1425 gives the remainder as 7 and 9 on dividing by 12.]

Now, the sum is reduced to,
$$\frac{{5 \times 7 \times 9}}{{12}} = \frac{{35 \times 9}}{{12}}$$
$$\frac{{35 \times 9}}{{12}}$$  = Remainder ⇒ -1 × -3 = 3 [Here, we have taken negative remainder.] So, required remainder will be 3.

Note: When, $$\frac{9}{{12}}$$ it gives positive remainder as 9 and it also give a negative remainder -3. As per our convenience,we can take any time positive or negative remainder.