Question

Let \[N=1421\times 1423\times 1425\]. What is the remainder when N is divided by 12?
(a) 0
(b) 9
(c) 3
(d) 6

Answer 1

Hint: In this question, let us first divide the three numbers separately by 12 and then get the remainder for them. Then multiply the three remainders so obtained and divide that again with 12. Now, the remainder obtained will be the result.

Complete step-by-step answer:
Now from the given N in the question.
\[N=1421\times 1423\times 1425\]
Let us first divide the number 1421 by 12.
Now, 1421 can be written as:
\[\Rightarrow 1421=118\times 12+5\]
Thus, the remainder in this case is 5.
Let us now divide the number 1423 by 12.
Now, the number 1423 can be written as:
\[\Rightarrow 1423=118\times 12+7\]
Thus, the remainder in this case is 7.
Let us now divide the number 1425 by 12.
Now, the number 1425 can be written as:
\[\Rightarrow 1425=118\times 12+9\]
Thus, the remainder when 1425 divided by 12 is 9.
Now, to get the remainder for N we have to multiply each of the remainders that are obtained in each case.
Now, the remainder of N will be:
\[\begin{align}
  & \Rightarrow 5\times 7\times 9 \\
 & \Rightarrow 315 \\
\end{align}\]
 Let us now again divide this remainder of N with 12.
Now, again 315 can be written as:
\[\Rightarrow 315=26\times 12+3\]
Now, when we divide it with 12 we get the remainder as 3.
Hence, the correct option is (c).

Note: It is important to note that we do not need to multiply the three numbers given in N and then divide it with 12. Instead we can divide each of the terms by 12 to get the remainder separately and then multiply all of them which is easy to calculate.
After getting the remainder to each of the terms in N we need to multiply them and then again need to find the remainder because the number so obtained can be again divided. Neglecting any of the terms in between changes the result completely.