Can two numbers have 18 as their HCF and 380 as their LCM?
To determine whether two numbers can have 18 as their highest common factor (HCF) and 380 as their least common multiple (LCM), we need to understand the relationship between HCF, LCM, and the numbers themselves.

Relationship between HCF, LCM, and Numbers
- The HCF of two numbers is the largest number that divides both numbers evenly.
- The LCM of two numbers is the smallest number that is divisible by both numbers.
- The relationship between two numbers, their HCF, and LCM is given by the product of the two numbers equals the product of their HCF and LCM. Mathematically, for two numbers a and b:
a * b = HCF(a, b) * LCM(a, b)

Analysis of the Given Numbers
- We are given that the HCF of the two numbers is 18, and the LCM is 380.
- Let the two numbers be x and y.
- Therefore, we have x * y = 18 * 380
x * y = 6840

Determining the Numbers
- To find two numbers whose product is 6840, we can list down the pairs of factors of 6840 and find a pair where the HCF is 18.
- The factors of 6840 are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 17, 20, 23, 30, 34, 40, 45, 46, 51, 60, 68, 69, 85, 90, 92, 102, 115, 138, 170, 180, 204, 230, 255, 276, 340, 345, 460, 510, 575, 690, 765, 920, 1020, 1150, 1380, 1530, 2300, 2760, 3060, 4600, 6120, and 6840.
- Observing the factors, we can find two numbers that have 18 as their HCF and 380 as their LCM, which are 90 and 76.
- Therefore, the numbers 90 and 76 can have 18 as their HCF and 380 as their LCM.
Therefore, it is possible for two numbers to have 18 as their HCF and 380 as their LCM, as demonstrated by the example of 90 and 76.