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 step by step, we can follow this method:

### Step 1: Define the number
Let the number be \( n \).

### Step 2: Use the division algorithm
According to the division algorithm, we can express \( n \) in terms of the divisor, quotient, and remainder:
\[ n = 143q + 31 \]
where \( q \) is some integer (the quotient when \( n \) is divided by \( 143 \)).

### Step 3: Find the remainder when \( n \) is divided by \( 13 \)
We want to find the remainder when \( n \) is divided by \( 13 \):
\[ n = 143q + 31 \]
We can substitute \( 143 \) in terms of \( 13 \):
\[ 143 = 13 \times 11 \]
So, we can rewrite \( n \):
\[ n = (13 \times 11)q + 31 \]

### Step 4: Simplify the expression
Now we can express \( n \) as:
\[ n = 13 \times (11q) + 31 \]
Next, we need to find \( 31 \) modulo \( 13 \):
\[ 31 \div 13 = 2 \quad \text{(quotient)} \]
Calculating the remainder:
\[ 31 - (13 \times 2) = 31 - 26 = 5 \]
Thus, \( 31 \equiv 5 \mod 13 \).

### Step 5: Combine the results
Now we can write:
\[ n = 13 \times (11q) + 31 \equiv 0 + 5 \mod 13 \]
This means that when \( n \) is divided by \( 13 \), the remainder is \( 5 \).

### Final Answer
The remainder when the number \( n \) is divided by \( 13 \) is \( 5 \).

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