A number when divided by 114, leaves remainder 21. If the same number is divided by 19, then the remainder is :

Question

Accepted Answer

To solve the problem, we need to determine the remainder when the given number is divided by 19. Let's break down the problem step by step:

1. **Understanding the Problem:**
   - We have a number $ N $ which, when divided by 114, leaves a remainder of 21. This can be expressed as:
     $$
     N = 114k + 21
     $$
     for some integer $ k $.

2. **Finding the Remainder when Divided by 19:**
   - We need to find the remainder when $ N $ is divided by 19. Substituting the expression for $ N $:
     $$
     N = 114k + 21
     $$
   - We are interested in $ N \mod 19 $:
     $$
     N \equiv 114k + 21 \pmod{19}
     $$

3. **Simplifying the Expression:**
   - First, find $ 114 \mod 19 $:
     $$
     114 \div 19 = 6 \quad \text{remainder} \quad 0 \quad \Rightarrow \quad 114 \equiv 0 \pmod{19}
     $$
   - Therefore, the expression simplifies to:
     $$
     N \equiv 0k + 21 \equiv 21 \pmod{19}
     $$

4. **Calculating $ 21 \mod 19 $:**
   - Now, calculate the remainder of 21 when divided by 19:
     $$
     21 \div 19 = 1 \quad \text{remainder} \quad 2 \quad \Rightarrow \quad 21 \equiv 2 \pmod{19}
     $$

5. **Conclusion:**
   - The remainder when the number $ N $ is divided by 19 is 2. Therefore, the correct answer is Option 3.

**Explanation of Incorrect Options:**

- **Option 1 (0):** This would imply that 21 is a multiple of 19, which it is not, as 21 divided by 19 leaves a remainder of 2.
- **Option 2 (1):** This would imply that 21 divided by 19 leaves a remainder of 1, which is incorrect because the remainder is 2.
- **Option 4 (5):** This would imply that 21 divided by 19 leaves a remainder of 5, which is incorrect because the remainder is 2.

Thus, the correct answer is indeed Option 3, as the remainder when the number is divided by 19 is 2.