Difficulty: Medium
Correct Answer: 36 and 60
Explanation:
Introduction:
This question combines the concepts of HCF and factor pairs of a product. You know the product of two numbers and their HCF and must determine the two original two digit numbers. This type of reasoning is common in aptitude exams.
Given Data / Assumptions:
Concept / Approach:
Let the two numbers be 12x and 12y, because 12 is their HCF. Then their HCF is 12 × HCF(x, y). For the HCF to be exactly 12, x and y must be co-prime. Also, the product 12x × 12y must equal 2160. This gives an equation in terms of x and y that we can solve to find co-prime factors.
Step-by-Step Solution:
Let the numbers be 12x and 12y. Their product is 12x × 12y = 144xy. We are given that 144xy = 2160. So xy = 2160 ÷ 144 Compute: 2160 ÷ 144 = 15 Thus x and y are co-prime factors of 15. The positive factor pairs of 15 are (1, 15) and (3, 5). If x = 1, y = 15, the numbers are 12 and 180 (but 180 is not a two digit number). If x = 3, y = 5, the numbers are 36 and 60, both two digit numbers. Therefore, the required numbers are 36 and 60
Verification / Alternative check:
Check product: 36 × 60 = 2160, which matches the given product. Check HCF(36, 60): 36 = 2^2 × 3^2, 60 = 2^2 × 3 × 5, so common primes with smallest powers give HCF = 2^2 × 3 = 12. Both conditions are satisfied.
Why Other Options Are Wrong:
60 and 72 have HCF 12 but their product is 4320, not 2160. 12 and 60 have product 720 and also 12 is not a two digit number. 72 and 30 have product 2160 but HCF(72, 30) is 6, not 12. Only 36 and 60 satisfy both the HCF and product conditions.
Common Pitfalls:
Students may only match the product 2160 or only match the HCF 12, forgetting that both conditions must simultaneously hold true. Others may ignore the two digit requirement, accidentally choosing numbers like 12 and 180, which would be invalid here.
Final Answer:
The two required two digit numbers are 36 and 60.
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