If the sum of squares of two numbers is 2754 and their HCF and LCM are 9 and…
2023
If the sum of squares of two numbers is 2754 and their HCF and LCM are 9 and 135 respectively, then the numbers are
- A.
27,36
- B.
28,45
- C.
27,35
- D.
27,45
Attempted by 20 students.
Show answer & explanation
Correct answer: D
Key idea: express the numbers in terms of their HCF and use the LCM to find the co-prime factors.
Let the two numbers be 9a and 9b, where gcd(a,b) = 1 (since 9 is the HCF).
Their LCM is 9ab. Given LCM = 135, so 9ab = 135 ⇒ ab = 15.
Find coprime factor pairs of 15. The pair of positive coprime factors is 3 and 5 (since 1 and 15 are not coprime to produce the required sum of squares later). Thus a = 3, b = 5 (or vice versa).
So the numbers are 9×3 = 27 and 9×5 = 45.
Verify conditions: 27^2 + 45^2 = 729 + 2025 = 2754; HCF(27,45) = 9; LCM(27,45) = 135. All given conditions are satisfied.
Answer: 27 and 45.