- A
- B
- C
- D
Step-by-Step Solution
and 455 − 420 = 35, so with 0 ≤ 35 < 42. The others break the lemma's condition 0 ≤ is negative, is larger than the divisor, and equals (one more 42 can still be taken out).
Answer: D)
Apply Euclid's division lemma to a = 455 and b = 42, writing a = bq + r. The correct pair (q, r) is:
Class 10Real NumbersNumber Systems
Key idea: Euclid's Division Lemma and HCF
**Lemma**: For positive integers a, (with ), there exist UNIQUE with and 0 ≤ .
This is just formal long division — is the quotient, is the remainder.
**Algorithm to find HCF(a, b)** with :
1. Apply lemma: .
2. If , HCF = .
3. If ≠ 0, replace (a, b) with , r) and repeat.
**Example**: HCF(420, 130)
→ HCF = 10.
This is just formal long division — is the quotient, is the remainder.
**Algorithm to find HCF(a, b)** with :
1. Apply lemma: .
2. If , HCF = .
3. If ≠ 0, replace (a, b) with , r) and repeat.
**Example**: HCF(420, 130)
→ HCF = 10.
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