If the LCM of two numbers is 180 and their HCF is 15, what is the sum of the two

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Question: If the LCM of two numbers is 180 and their HCF is 15, what is the sum of the two numbers? (2023)

Options:

Correct Answer: 105

Exam Year: 2023

Solution:

Let the two numbers be 15a and 15b. Then, LCM = 15ab = 180, so ab = 12. The pairs (3, 4) give the numbers 45 and 60, summing to 105.

If the LCM of two numbers is 180 and their HCF is 15, what is the sum of the two numbers? (2023)

If the LCM of two numbers is 180 and their HCF is 15, what is the sum of the two numbers? (2023)

Step 1: Understand that LCM (Least Common Multiple) and HCF (Highest Common Factor) are related to two numbers.
Step 2: Let the two numbers be represented as 15a and 15b, where 15 is their HCF.
Step 3: Since the LCM of the two numbers is given as 180, we can use the formula: LCM = HCF * (a * b).
Step 4: Substitute the known values into the formula: 180 = 15 * (a * b).
Step 5: Simplify the equation: a * b = 180 / 15, which gives a * b = 12.
Step 6: Now, find pairs of numbers (a, b) that multiply to 12. The pairs are (1, 12), (2, 6), (3, 4).
Step 7: Calculate the actual numbers using the pairs: For (3, 4), the numbers are 15*3 = 45 and 15*4 = 60.
Step 8: Add the two numbers together: 45 + 60 = 105.

LCM and HCF Relationship – The relationship between the Least Common Multiple (LCM) and Highest Common Factor (HCF) of two numbers, which states that the product of the two numbers is equal to the product of their LCM and HCF.
Factorization – Understanding how to express numbers in terms of their factors, particularly when using HCF to simplify the problem.
Pairs of Factors – Identifying pairs of factors that satisfy a given product, which is essential for finding the original numbers.