Mathematics — Standard (041) — CBSE Class 10 board question

Prime Factorisation:

$$18180 = 2^2 \times 3 \times 5 \times 5 \times 101 = 2^2 \times 3 \times 5^2 \times 101$$

$$7575 = 3 \times 5 \times 5 \times 101 = 3 \times 5^2 \times 101$$

HCF = Product of smallest powers of common prime factors

$$\text{HCF}(18180,\ 7575) = 3^1 \times 5^2 \times 101 = 3 \times 25 \times 101 = \mathbf{7575}$$

LCM = Product of greatest powers of all prime factors

$$\text{LCM}(18180,\ 7575) = 2^2 \times 3 \times 5^2 \times 101 = 4 \times 3 \times 25 \times 101 = \mathbf{30300}$$

Source: Chapter 1, Section 1.2 — The Fundamental Theorem of Arithmetic

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• Key method: Prime factorise both numbers first, then apply the rules: HCF uses the lowest powers of common factors; LCM uses the highest powers of all factors.
• Notice here that 7575 divides 18180 exactly, so HCF = 7575 and LCM = 18180 × 2 = 30300. You can verify: HCF × LCM = 7575 × 30300 = 18180 × 7575 ✓
• Show all factorisation steps clearly — examiners award marks for working, not just the final answer.