Can two numbers have 18 as their HCF and 380 as their LCM? Give reasons.
No, such numbers cannot exist because \(18\) does not divide \(380\).
Step 1: Recall the property of HCF and LCM.
For any two positive integers \(a\) and \(b\):
\( \text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b \).
From this property, it follows that the HCF of two numbers must always divide their LCM.
Step 2: Check if the given HCF divides the given LCM.
Given HCF = \(18\), LCM = \(380\).
Check \(\dfrac{380}{18}\).
\(380 \div 18 = 21.11...\) (not an integer).
Step 3: Reasoning.
Since \(18\) does not divide \(380\), the pair of numbers cannot exist.
Conclusion: No two numbers can have HCF = \(18\) and LCM = \(380\).