A number when divided by 258 gives a remainder equal to 87. Find the remainder when the same number is divided by 43.Options0513

1. Break Down the Problem

We are given a number $N$ which when divided by 258 leaves a remainder of 87. Our task is to find the remainder of $N$ when divided by 43. We can express the relationship as: $N = 258k + 87$ for some integer $k$.

2. Relevant Concepts

To find the remainder of $N$ when divided by 43, we can rewrite the expression: $N \mod 43 = (258k + 87) \mod 43$ Using properties of modular arithmetic: $(a + b) \mod m = [(a \mod m) + (b \mod m)] \mod m$

3. Analysis and Detail

First, we compute $258 \mod 43$: $258 \div 43 \approx 6.0 \Rightarrow 258 = 43 \times 6 + r$ Calculating: $43 \times 6 = 258 \Rightarrow r = 0 \implies 258 \mod 43 = 0$ Next, compute $87 \mod 43$: $87 \div 43 \approx 2.0 \Rightarrow 87 = 43 \times 2 + r$ Calculating: $43 \times 2 = 86 \Rightarrow r = 1 \implies 87 \mod 43 = 1$

Now, substituting back into the equation: $N \mod 43 = (0 \cdot k + 1) \mod 43 = 1$

4. Verify and Summarize

We've calculated the respective remainders and combined them to find that the remainder when $N$ is divided by 43 is: $\text{Remainder} = 1$

Final Answer

The remainder when the same number is divided by 43 is 1.