A number when divided by `143` leaves `31` as remainder. What will be the remainder, when the same number is divided by `13` ?

Answer

To solve the problem, we need to find the remainder when a number that leaves a remainder of 31 when divided by 143 is divided by 13.

Let's denote the number as \( N \).

1. **Understanding the given information**:
We know that when \( N \) is divided by \( 143 \), it leaves a remainder of \( 31 \). This can be expressed mathematically as:
\[ N = 143k + 31 \]
for some integer \( k \).

2. **Finding the remainder when divided by 13**:
We need to find \( N \mod 13 \). We can substitute the expression for \( N \) into this equation:
\[ N \mod 13 = (143k + 31) \mod 13 \]

3. **Calculating \( 143 \mod 13 \)**:
First, we need to find \( 143 \mod 13 \):
\[ 143 \div 13 = 11 \quad \text{(since \( 11 \times 13 = 143 \))} \]
Therefore,
\[ 143 \mod 13 = 0 \]

4. **Substituting back**:
Now we can simplify our equation:
\[ N \mod 13 = (0 \cdot k + 31) \mod 13 = 31 \mod 13 \]

5. **Calculating \( 31 \mod 13 \)**:
Next, we find \( 31 \mod 13 \):
\[ 31 \div 13 = 2 \quad \text{(since \( 2 \times 13 = 26 \))} \]
The remainder is:
\[ 31 - 26 = 5 \]
Thus,
\[ 31 \mod 13 = 5 \]

6. **Final answer**:
Therefore, the remainder when the number \( N \) is divided by \( 13 \) is:
\[ \boxed{5} \]