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

Let the two numbers be $2x$ and $3x$ (since ratio is $2:3$).

We know: HCF × LCM = Product of the two numbers

$$\text{HCF} \times 180 = 2x \times 3x$$

Also, for numbers $2x$ and $3x$, HCF = $x$ (since 2 and 3 are coprime).

So: $x \times 180 = 2x \times 3x = 6x^2$

$$180 = 6x \implies x = 30$$

∴ HCF = 30

Source: Chapter 1, Section 1.2 (Fundamental Theorem of Arithmetic)

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• The key relation used is HCF × LCM = Product of two numbers — this is stated explicitly in the chapter.
• Since the ratio is $2:3$ (coprime integers), the HCF of $2x$ and $3x$ is simply $x$. Examiners expect students to state this clearly.
• Show all steps: define variables → apply the formula → solve for $x$ → state HCF. Don't skip steps in a 2-mark question.