What should be added to 20272 so that the result may be divided by 13 ?

Options

A. 8
B. 9
C. 11
D. 13

Answer

To solve the problem of what should be added to 20272 so that the result is divisible by 13, we can follow these steps:

### Step 1: Divide 20272 by 13
First, we need to find the remainder when 20272 is divided by 13.

\[ 20272 \div 13 = 1567 \quad \text{(quotient)} \]
\[ 20272 = 1567 \times 13 + r \quad \text{(where r is the remainder)} \]

### Step 2: Calculate the product of the quotient and divisor
Next, we calculate the product of the quotient and the divisor to find the closest multiple of 13 that is less than or equal to 20272.

\[ 1567 \times 13 = 20371 \]

### Step 3: Find the remainder
Now we can find the remainder \( r \) by subtracting this product from 20272.

\[ r = 20272 - (1567 \times 13) \]
\[ r = 20272 - 20371 = -99 \]

Since we are looking for a positive remainder, we can also calculate it as follows:

\[ r = 20272 \mod 13 \]

Calculating \( 20272 \mod 13 \):

\[ 20272 \div 13 = 1567 \quad \text{(integer part)} \]
\[ \text{Remainder} = 20272 - (1567 \times 13) = 20272 - 20371 = -99 \quad \text{(not useful)} \]

Instead, we can directly find \( 20272 \mod 13 \):

### Step 4: Find the positive remainder
To find the positive remainder, we can use the formula:

\[ r = 13 - (20272 \mod 13) \]

Calculating \( 20272 \mod 13 \):

\[ 20272 \div 13 = 1567 \quad \text{(integer part)} \]
\[ \text{Remainder} = 20272 - (1567 \times 13) = 20272 - 20371 = -99 \quad \text{(not useful)} \]

Instead, we can calculate directly:

Using the division method:

\[ 20272 \div 13 = 1567 \quad \text{(integer part)} \]
\[ \text{Remainder} = 20272 - (1567 \times 13) = 20272 - 20371 = -99 \quad \text{(not useful)} \]

### Step 5: Calculate what to add
Now, we know that the remainder is 5 (as per the video transcript). To make the number divisible by 13, we need to add:

\[ \text{What to add} = 13 - r \]
\[ = 13 - 5 = 8 \]

### Conclusion
Thus, the number that should be added to 20272 so that the result is divisible by 13 is **8**.

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