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Two pieces of cloth measure 96 metres and 144 metres in length. What is the maximum length of cloth that can be cut equally from both pieces without any remainder?
96 metres
24 metres
12 metres
48 metres
48 metres
To find the maximum length of cloth that can be cut equally from both pieces (96 metres and 144 metres) without any remainder, we need to calculate the Highest Common Factor (HCF) of 96 and 144. The prime factorizations are 96 = 2 to the power 5 times 3 and 144 = 2 to the power 4 times 3 squared, yielding an HCF of 48 metres.
To find the maximum length of cloth that can be cut equally from both pieces (96 metres and 144 metres) without any remainder, we need to calculate the Highest Common Factor (HCF) of 96 and 144. The prime factorizations are 96 = 2 to the power 5 times 3 and 144 = 2 to the power 4 times 3 squared, yielding an HCF of 48 metres.
The concept of Highest Common Factor (HCF) and Lowest Common Multiple (LCM) forms the fundamental basis of number theory in quantitative aptitude, originating from ancient Indian mathematical traditions found in the works of Aryabhata and Brahmagupta.
HCF represents the greatest divisor that divides two or more numbers without leaving a remainder.
Prime factorization of 96 yields 32 ├Ч 3 = 96, and 144 yields 16 ├Ч 9 = 144.
The common prime factors with the lowest exponents are 2 to the power 4 and 3 to the power 1, product being 48.
Option A (96 metres) cannot divide 144 without a remainder (144 ├╖ 96 leaves a remainder of 48).
Option B (24 metres) is a common factor but not the maximum possible length since 48 is higher.
Option C (12 metres) is a common factor, but it is not the highest common factor.
D is correct тАФ 48 metres is the maximum length of cloth that can be cut equally from both 96m and 144m pieces without remainder.
Mastering HCF and LCM word problems is essential for tackling time-and-work circular track problems and synchronization of traffic lights in competitive examinations.
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