Numbers from HCF and ratio: The HCF of three numbers is 23. If they are in the ratio 1 : 2 : 3, find the actual numbers.
Aptitude
Problems on H.C.F and L.C.M
Difficulty: Easy
Choose an option
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A69, 15, 22
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B23, 46, 69
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C25, 31, 41
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D23, 21, 35
Answer
Correct Answer: 23, 46, 69
Explanation
Introduction / Context:When numbers are in a given ratio and share a known HCF, each number can be expressed as HCF times the respective ratio part (provided the ratio parts are coprime in aggregate). This is a standard reconstruction technique in HCF–ratio problems.
Given Data / Assumptions:
- HCF = 23
- Ratio = 1 : 2 : 3
Concept / Approach:Let the numbers be 23x, 23y, 23z with x : y : z = 1 : 2 : 3. Because the common factor extracted is the HCF, the simplest choice is x = 1, y = 2, z = 3 to obtain the minimal triple satisfying the ratio and HCF conditions.
Step-by-Step Solution:
Number 1 = 23 * 1 = 23.Number 2 = 23 * 2 = 46.Number 3 = 23 * 3 = 69.Verification / Alternative check:
HCF(23, 46, 69) = 23, and 23 : 46 : 69 reduces to 1 : 2 : 3 as required.Why Other Options Are Wrong:
- All other triplets either do not have HCF equal to 23 or do not preserve the exact 1 : 2 : 3 ratio.
Common Pitfalls:
- Multiplying the ratio parts by another common factor, which would increase the HCF beyond 23.
Final Answer:
23, 46, 69