For two positive integers, HCF × LCM equals:

The relationship between the Highest Common Factor (HCF) and the Least Common Multiple (LCM) of two positive integers $a$ and $b$ is defined by a fundamental mathematical identity. ### Formula: For any two positive integers $a$ and $b$: $$HCF(a, b) \times LCM(a, b) = a \times b$$ ### Step-by-Step Breakdown: 1. **Identify the Terms**: Let the two numbers be $a$ and $b$. - **HCF** (Highest Common Factor) is the largest positive integer that divides each of the integers. - **LCM** (Least Common Multiple) is the smallest positive integer that is divisible by both numbers. 2. **Verification with an Example**: Let $a = 12$ and $b = 18$. - Prime factorization of $12 = 2^2 \times 3$ - Prime factorization of $18 = 2 \times 3^2$ - $HCF(12, 18) = 2^1 \times 3^1 = 6$ - $LCM(12, 18) = 2^2 \times 3^2 = 36$ 3. **Calculation**: - $HCF \times LCM = 6 \times 36 = 216$ - Product of the numbers ($a \times b$) = $12 \times 18 = 216$ 4. **Conclusion**: Since $216 = 216$, the property is verified. The product of the HCF and LCM of two numbers is always equal to the **product of the numbers**.