The correct answer is option (B):
2
Let's break down this modular arithmetic problem step by step.
We are given that when an integer 'n' is divided by 'k', the remainder is 1. This can be expressed as:
n ≡ 1 (mod k) ...Equation 1
This means that n can be written in the form: n = q1 * k + 1, where q1 is some integer.
We are also given that when another integer 'm' is divided by 'k', the remainder is 2. This can be expressed as:
m ≡ 2 (mod k) ...Equation 2
This means that m can be written in the form: m = q2 * k + 2, where q2 is some integer.
Now, we want to find the remainder when 'n x m' is divided by 'k'. Let's multiply the congruences from Equation 1 and Equation 2. Remember that when working with congruences, we can multiply them just like regular equations.
n * m ≡ 1 * 2 (mod k)
n * m ≡ 2 (mod k)
This tells us that when n * m is divided by k, the remainder is 2.
Alternatively, we can multiply the expressions for n and m:
n * m = (q1 * k + 1) * (q2 * k + 2)
n * m = q1 * q2 * k^2 + 2 * q1 * k + q2 * k + 2
n * m = k * (q1 * q2 * k + 2 * q1 + q2) + 2
Notice the first term is a multiple of k. The second term is a 2. Therefore when n*m is divided by k, the remainder is 2.
Therefore, the correct answer is 2.