If the HCF of two numbers is 5 and their LCM is 100, what is the sum of the two numbers? (2023)
Practice Questions
1 question
Q1
If the HCF of two numbers is 5 and their LCM is 100, what is the sum of the two numbers? (2023)
30
25
50
45
Let the two numbers be 5x and 5y. Then, HCF(5x, 5y) = 5 and LCM(5x, 5y) = 100. This gives xy = 20. The pairs (x, y) that satisfy this are (4, 5) or (5, 4), leading to the sum 5(4 + 5) = 45.
Questions & Step-by-step Solutions
1 item
Q
Q: If the HCF of two numbers is 5 and their LCM is 100, what is the sum of the two numbers? (2023)
Solution: Let the two numbers be 5x and 5y. Then, HCF(5x, 5y) = 5 and LCM(5x, 5y) = 100. This gives xy = 20. The pairs (x, y) that satisfy this are (4, 5) or (5, 4), leading to the sum 5(4 + 5) = 45.
Steps: 11
Step 1: Understand that HCF (Highest Common Factor) of two numbers is 5 and LCM (Lowest Common Multiple) is 100.
Step 2: Let the two numbers be represented as 5x and 5y, where x and y are some integers.
Step 3: Since the HCF of 5x and 5y is 5, it means that x and y must be coprime (they have no common factors other than 1).
Step 4: The formula relating HCF and LCM is: HCF(a, b) * LCM(a, b) = a * b. Here, a = 5x and b = 5y.
Step 5: Substitute the values into the formula: 5 * 100 = (5x) * (5y).
Step 6: Simplify the equation: 500 = 25xy.
Step 7: Divide both sides by 25: xy = 20.
Step 8: Now, find pairs of integers (x, y) that multiply to 20. The pairs are (1, 20), (2, 10), (4, 5), (5, 4), (10, 2), and (20, 1).
Step 9: Since x and y must be coprime, the valid pairs are (4, 5) and (5, 4).
Step 10: Calculate the two numbers: For (4, 5), the numbers are 5*4 = 20 and 5*5 = 25. For (5, 4), the numbers are 25 and 20 (same numbers).
