Question

If you divide the number 1003 by 5, then the remainder is equal to:

• A. 2
• B. 4
• C. 1
• D. 3

Hint:

To instantly find the remainder of any massive integer divided by 5, look exclusively at the final digit: - If the last digit is $\lt 5$, the remainder is the last digit itself. - If the last digit is $\ge 5$, the remainder is $(\text{last digit} - 5)$. For 1003, the last digit is 3, which is less than 5. Hence, the remainder is immediately 3.

Answer 1

The Correct Option is D

Concept: According to the Euclidean Division Lemma, any positive integer $a$ can be uniquely expressed in terms of a divisor $b$ such that $a = bq + r$, where $q$ is the quotient and $r$ is the remainder satisfying the inequality constraint $0 \le r \lt b$. For a divisor $b = 5$, the only permissible remainders are $r \in \{0, 1, 2, 3, 4\}$.

Step-by-step Explanation: Method A: Direct Long Division Approach
Let us execute the physical division operation of the dividend 1003 by the divisor 5:

• Divide the first two digits (10) by 5: $10 \div 5 = 2$, with a remainder of 0.

• Bring down the next digit (0): $0 \div 5 = 0$, with a remainder of 0.

• Bring down the final units digit (3): $3 \div 5 = 0$, with a remainder of 3.
Thus, the division gives a whole quotient of 200 with an un-divisible leftover value of 3. Mathematically, this is framed as: $$1003 = 5 \times 200 + 3$$ Since $3$ satisfies $0 \le 3 \lt 5$, the remainder is strictly equal to 3. Method B: Using Number Theory and Modular Divisibility Rules
Under modular arithmetic rules, a number's remainder modulo 5 depends entirely on its units digit. A positive integer is perfectly divisible by 5 if and only if its rightmost digit is either 0 or 5. Consequently, we can state that: $$a \equiv (\text{last digit of } a) \pmod 5$$ The number given is 1003. Its last digit is 3. Applying the definition: $$1003 \equiv 3 \pmod 5$$ Since 3 is less than 5, the remainder when 1003 is divided by 5 is 3.