Question

For questions 6-7, Two statements are given - one labelled Assertion (A) and the other labelled Reason (R).

Select the correct answer to these questions from the codes (a), (b), (c) and (d) as given below:

a) Both (A) and (R) are true and (R) is the correct explanation of (A)

b) Both (A) and (R) are true but (R) is not the correct explanation of (A)

c) (A) is true but (R) is false

d) (A) is false but (R) is true

Assertion (A): When $17^{113}$ is divided by 3, the remainder is 2.

Reason (R): $17 \equiv -1 \pmod{3}$ so $17^{113} \equiv (-1)^{113} \pmod{3}$.

• A. Both (A) and (R) are true and (R) is the correct explanation of (A)
• B. Both (A) and (R) are true but (R) is not the correct explanation of (A)
• C. (A) is true but (R) is false
• D. (A) is false but (R) is true

Answer 1

Answer: (A)

a) Both (A) and (R) are true and (R) is the correct explanation of (A)

Answer 2

Detailed analysis of the Assertion and Reason:

Assertion (A): When $17^{113}$ is divided by 3, the remainder is 2.

To verify this, we need to find the value of $17^{113} \pmod{3}$.

First, find the remainder of 17 when divided by 3: $17 = 5 \times 3 + 2$ So, $17 \equiv 2 \pmod{3}$.

Now, we can substitute this into the expression: $17^{113} \equiv 2^{113} \pmod{3}$.

To simplify $2^{113} \pmod{3}$, we can observe the pattern of powers of 2 modulo 3: $2^1 \equiv 2 \pmod{3}$ $2^2 \equiv 4 \equiv 1 \pmod{3}$ $2^3 \equiv 2^2 \cdot 2^1 \equiv 1 \cdot 2 \equiv 2 \pmod{3}$ $2^4 \equiv 2^2 \cdot 2^2 \equiv 1 \cdot 1 \equiv 1 \pmod{3}$

The pattern is $2, 1, 2, 1, \dots$. If the exponent is odd, the remainder is 2. If the exponent is even, the remainder is 1.

Since 113 is an odd number, $2^{113} \equiv 2 \pmod{3}$. Therefore, $17^{113} \equiv 2 \pmod{3}$.

This means when $17^{113}$ is divided by 3, the remainder is 2. Thus, Assertion (A) is True.

Reason (R): $17 \equiv -1 \pmod{3}$ so $17^{113} \equiv (-1)^{113} \pmod{3}$.

Let's check the first part of the reason: $17 \equiv -1 \pmod{3}$. We know $17 = 5 \times 3 + 2$, so $17 \equiv 2 \pmod{3}$. Also, $2 \equiv -1 \pmod{3}$ because $2 - (-1) = 3$, which is a multiple of 3. So, $17 \equiv 2 \equiv -1 \pmod{3}$. This part is True.

Now, let's check the second part: $17^{113} \equiv (-1)^{113} \pmod{3}$. This is a direct application of the property of modular arithmetic: If $a \equiv b \pmod{n}$, then $a^k \equiv b^k \pmod{n}$ for any positive integer $k$. Since $17 \equiv -1 \pmod{3}$, it follows that $17^{113} \equiv (-1)^{113} \pmod{3}$. This part is also True. Thus, Reason (R) is True.

Evaluating if Reason (R) is the correct explanation of Assertion (A):

Reason (R) provides a method to calculate $17^{113} \pmod{3}$. It states $17 \equiv -1 \pmod{3}$. Then, $17^{113} \equiv (-1)^{113} \pmod{3}$. Since 113 is an odd number, $(-1)^{113} = -1$. So, $17^{113} \equiv -1 \pmod{3}$. To express the remainder as a non-negative integer less than the divisor (3), we add 3 to -1: $-1 + 3 = 2$. Therefore, $17^{113} \equiv 2 \pmod{3}$.

This calculation directly leads to the result stated in Assertion (A). The reason clearly explains the steps and properties used to arrive at the remainder. Hence, Reason (R) is the correct explanation for Assertion (A).

Conclusion: Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).