Q.The least non-negative remainder, when $3^{15}$ is divided by 7 is :

Remainder Division: From Sharing to a Precise Rule

Think about what happens when you try to share things equally. Suppose you have 17 marbles and 5 friends. You give each friend 3 marbles — that uses 15 marbles. You have 2 marbles left over that you cannot share equally among 5 friends without breaking them.

That leftover is the remainder. The whole process is remainder division.

The intuition is simple: when you divide a number by another, you are asking two questions at once:

• How many full groups can I make? (the quotient)
• How many are left over, too few to make another full group? (the remainder)

The Precise Statement

For any two whole numbers $a$ (the dividend) and $b$ (the divisor, $b \neq 0$), there exist unique whole numbers $q$ (the quotient) and $r$ (the remainder) such that:

with the condition that $0 \leq r < b$.

That last inequality is the key rule: the remainder must always be smaller than the divisor. If it is not, you could have made another full group.

How It Works in Practice

Take $a = 17$, $b = 5$.

You ask: what is the largest whole number $q$ such that $5 \times q \leq 17$? That is $q = 3$, because $5 \times 3 = 15$ and $5 \times 4 = 20$ (too big). Then the remainder is $r = 17 - 15 = 2$. Check: $0 \leq 2 < 5$ — satisfied.

So $17 = 5 \times 3 + 2$.

Why the Condition $0 \leq r < b$ Matters

Without it, you could write $17 = 5 \times 2 + 7$ — that is mathematically true, but it is useless because the "remainder" 7 is larger than the divisor 5. You could have made one more full group of 5. The condition forces the remainder to be the smallest possible non-negative leftover, which makes the representation unique.

The Division Algorithm Statement

This is often called the Division Algorithm (though it is a theorem, not an algorithm in the programming sense):

For every pair of integers $a$ and $b$ ($b > 0$), the integers $q$ and $r$ exist and are unique.

A Quick Table of Examples

Notice the last row: when the dividend is smaller than the divisor, the quotient is 0 and the remainder is the dividend itself. That is perfectly valid.

Why This Matters

Remainder division is the foundation for:

• Checking divisibility (remainder 0 means $b$ divides $a$ exactly)
• Finding GCD using Euclid's algorithm
• Modular arithmetic (clock arithmetic)
• Number theory in general

Once you internalize $a = bq + r$ with $0 \leq r < b$, you have the precise language for everything from simple sharing to advanced mathematics.