Divide the largest 5 digit number by 653. Check your answer by the division algorithm.

Answer

To solve the problem of dividing the largest 5-digit number by 653 and checking the answer using the division algorithm, follow these steps:

### Step 1: Identify the largest 5-digit number
The largest 5-digit number is 99999.

### Step 2: Perform the division
Now, we need to divide 99999 by 653.

\[ 99999 \div 653 \]

### Step 3: Calculate the quotient and remainder
1. **Perform the division**:
- 653 goes into 99999 a certain number of times. We can estimate this by calculating how many times 653 fits into 99999.
- Using long division, we find that 653 fits into 99999 approximately 153 times.

2. **Calculate the product**:
- Multiply the quotient (153) by the divisor (653) to find the product:
\[ 153 \times 653 = 99909 \]

3. **Calculate the remainder**:
- Now, subtract the product from the original number:
\[ 99999 - 99909 = 90 \]

So, we have:
- Quotient = 153
- Remainder = 90

### Step 4: Write the final answer
The division can be expressed as:
\[ 99999 = 653 \times 153 + 90 \]

### Step 5: Check using the division algorithm
According to the division algorithm, for any integers \( a \) (dividend), \( b \) (divisor), there exist unique integers \( q \) (quotient) and \( r \) (remainder) such that:
\[ a = b \times q + r \]
where \( 0 \leq r < b \).

In our case:
- \( a = 99999 \)
- \( b = 653 \)
- \( q = 153 \)
- \( r = 90 \)

Since \( 0 \leq 90 < 653 \), the division algorithm holds true.

### Final Result
The quotient is **153** and the remainder is **90**.

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