Euclid's division lemma — Class 10 CBSE Mathematics MCQs with Solutions

Solution: Euclid's division lemma states that for any two positive integers $a$ and $b$, we can express $a = bq + r$ where $q$ is the quotient and $r$ is the remainder. The crucial condition is that the remainder must satisfy $0 \leq r < b$ — it cannot be negative and must be strictly less than the divisor. This form is unique for the given $a$ and $b$.

Solution: When $r = 0$, the equation becomes $a = bq + 0 = bq$. This means $a$ is exactly divisible by $b$, or in other words, $b$ is a divisor of $a$. The remainder being zero is the defining condition for divisibility.

Solution: The constraint $0 \leq r < b$ guarantees uniqueness. Without this restriction, multiple pairs of $(q, r)$ could satisfy $a = bq + r$. For example, $10 = 3 \times 3 + 1$ and also $10 = 3 \times 2 + 4$ both work mathematically, but only the first respects the constraint. This uniqueness is what makes the lemma powerful and useful for mathematical proofs.

Solution: By definition of Euclid's division lemma, $q$ represents the quotient — the number of times $b$ goes into $a$ — and $r$ represents the remainder. These are uniquely determined by the condition $0 \leq r < b$. The statement is always true because this is precisely what $q$ means in the context of the lemma.

Solution: When 47 pencils are divided among 6 students, we apply Euclid's division lemma: $47 = 6 \times 7 + 5$. Each of the 6 students gets 7 pencils (the quotient), and 5 pencils remain (the remainder $r = 5$). The remainder is the leftover quantity, and it satisfies $0 \leq 5 < 6$. This is a real-world interpretation of the lemma.

Solution: Uniqueness is critical because it means Euclid's division lemma gives us exactly one answer every time. This allows us to build algorithms that always work correctly (like the Euclidean algorithm for GCD), and it provides a rigorous foundation for mathematical proofs. Without uniqueness, we couldn't rely on the lemma for logical deduction.

Solution: Euclid's division lemma can be extended to negative integers while maintaining the constraint $0 \leq r < |b|$. For $-17$ and $5$: we express $-17 = 5 \times (-4) + 3$. Notice that $q = -4$ (not $-3$) so that the remainder stays non-negative. This extension preserves uniqueness and the division property, making it a powerful tool even for negative integers.

Solution: Uniqueness means that once we choose specific values for $a$ and $b$, there is only one pair $(q, r)$ that can satisfy $a = bq + r$ with $0 \leq r < b$. We cannot find two different valid pairs. This is what makes the lemma a fundamental theorem — it guarantees that the division process has a predictable, single outcome.