Condition on r in Euclid’s Division Lemma

Video Explanation

Watch the video below for a clear explanation:

Solution

Question: Euclid’s division lemma states that for two positive integers a and b, there exist unique integers q and r such that

a = bq + r

where r must satisfy:

Euclid’s Division Lemma Condition

According to Euclid’s division lemma, the remainder r must satisfy the condition:

0 ≤ r < b

Explanation

• The remainder r cannot be negative.
• The remainder r must always be less than the divisor b.
• If r were equal to or greater than b, the division would not be complete.

Final Answer

✔ r must satisfy 0 ≤ r < b

Conclusion

Thus, Euclid’s division lemma guarantees that for any two positive integers, the remainder is always non-negative and strictly less than the divisor.