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A student claims that for three positive integers $p,q,r$, we always have $\operatorname{HCF}(p,q,r)\times \operatorname{LCM}(p,q,r)=p\times q\times r$. Which option is correct?
AThe claim is always true for any three positive integers.
BThe claim is true only when all three numbers are prime.
CThe claim is false in general for three numbers.
DThe claim is true only when two of the numbers are equal.
Answer & Solution
Correct answer: C. The claim is false in general for three numbers.
For two positive integers, $\operatorname{HCF}\times \operatorname{LCM}=\text{product of the numbers}$. But the chapter explicitly notes that this does not generally extend to three numbers. For example, for $6,72,120$, we get $\operatorname{HCF}=6$ and $\operatorname{LCM}=360$, so $6\times 360\neq 6\times 72\times 120$.
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