Explanation
Answer
To find the quotient and remainder when dividing 75578 by 53, we perform the division:
$\begin{align*}
75578 \div 53 & = \text{Quotient} \\
\text{Remainder} & = \text{0} \\
\end{align*}$
Breaking it down:

Divide the numbers:
$\begin{array}{r}
75578 \div 53 = 1426 \\
\\
\text{1426} \times \text{53} = 75578
\end{array}$

Calculate the remainder:
Since $75578 \mod 53 = 0$:
$\text{Remainder} = 0$
Thus, when 75578 divided by 53, the quotient is 1426 and the remainder is 0.
Long Division
Understanding Long Division
Long division is a method used to divide larger numbers by breaking them down into simpler steps. It is particularly useful when dealing with multidigit numbers. Here, we'll break down the process into a clear, stepbystep explanation.
Steps Involved
 Divide the first digit of the dividend by the divisor.
 Multiply the result by the divisor and write the product beneath the dividend.
 Subtract this product from the current number to find the remainder.
 Bring down the next digit and append it to the remainder.
 Repeat the process until all digits have been used.
Example
Let's divide 843 by 5 using long division.
 Divide: How many times does 5 go into 8? It goes 1 time.
 Multiply: $1 \times 5 = 5$
 Subtract: $8  5 = 3$
 Bring down: Bring down the 4, making it 34.
Repeat:
 Divide: How many times does 5 go into 34? It goes 6 times.
 Multiply: $6 \times 5 = 30$
 Subtract: $34  30 = 4$
 Bring down: Bring down the 3, making it 43.
Repeat:
 Divide: How many times does 5 go into 43? It goes 8 times.
 Multiply: $8 \times 5 = 40$
 Subtract: $43  40 = 3$
So, the result of 843 divided by 5 is 168 with a remainder of 3.
Mathematical Representation
In mathematical terms, long division can be represented as follows:
$\begin{array}{rl}
\phantom{88}843 & 5 \\ \hline
& 168 \\
 \phantom{88}5 & \\
\hline
\phantom{8}34 & \\
 \phantom{8}30 & \\
\hline
\phantom{8}\ 43 & \\
 \phantom{8}\ 40 & \\
\hline
\phantom{8}\ \phantom{8}\ 3 & \\
\end{array}$
As seen in this illustration, the numbers align under each other, keeping the process organized and clear.
Important Notes
 Always align each digit of the dividend and the quotient properly.
 After each subtraction, the result must be less than the divisor before bringing down the next digit.
 Practice is crucial to mastering long division; the more problems you solve, the more comfortable you'll become with the process.
This method provides a systematic way to handle large numbers, making it an essential tool in arithmetic.
Quotient And Remainder Calculation
Explanation
The concept of quotient and remainder calculation is fundamental in arithmetic and number theory. When you divide one integer by another, the quotient represents how many times the divisor fits into the dividend, while the remainder is what's left over after accounting for whole number divisions.
Mathematically, if you have two integers, $a$ (dividend) and $b$ (divisor), the division operation can be represented as:
$a = bq + r$
where:
 $q$ is the quotient
 $r$ is the remainder
Important Properties

The remainder $r$ must be nonnegative and less than the divisor $b$:
$0 \leq r < b$

The relationship can be derived from the division equation:
$q = \left\lfloor \frac{a}{b} \right\rfloor$
$r = a  bq$
Example
Consider dividing 17 by 5.
 Quotient: How many times does 5 fit into 17? It fits 3 times, so $q = 3$.
 Remainder: What is left after accounting for the division? From the equation $17 = 5 \cdot 3 + r$, we get $r = 2$.
Thus, the division $17 \div 5$ yields a quotient of 3 and a remainder of 2, which can be written as:
$17 = 5 \cdot 3 + 2$
Understanding these calculations is essential for various mathematical applications, including algorithms, modular arithmetic, and more.