Can two numbers have 16 as their Highest Common Factor (HCF) and 380 as their Lowest Common Multiple (LCM)?
To determine if two numbers can have 16 as their HCF and 380 as their LCM, we need to understand the relationship between HCF and LCM.

Relationship between HCF and LCM:
- For two numbers A and B, HCF(A, B) * LCM(A, B) = A * B

Given values:
- HCF = 16
- LCM = 380

Calculating the product:
- 16 * 380 = 6080

Explanation:
- If two numbers have 16 as their HCF, it means they have common factors of 2 and 8 (since 16 = 2^4)
- If the LCM of the two numbers is 380, it means the numbers must have factors of 2, 5, and 19 (since 380 = 2 * 2 * 5 * 19)
- However, the product of the two numbers must be 6080 to satisfy the relationship between HCF and LCM
- Since the prime factorization of 6080 is 2^6 * 5 * 19, the two numbers must have these factors in common
- Therefore, it is not possible for two numbers to have 16 as their HCF and 380 as their LCM simultaneously

Conclusion:
- Two numbers cannot have 16 as their HCF and 380 as their LCM at the same time due to the conflicting prime factors required to satisfy both conditions.