For any natural numbers \(m\), \(n\), and \(k\), such that \(k\) divides both \(m + 2n\) and \(3m + 4n\), \(k\) must be a common divisor of

Question

Sam can complete a job in 20 days when working alone. Mohit is twice as fast as Sam and thrice as fast as Ayana in the same job. They undertake a job with an arrangement where Sam and Mohit work together on the first day, Sam and Ayana on the second day, Mohit and Ayana on the third day, and this three-day pattern is repeated till the work gets completed. Then, the fraction of total work done by Sam is

Answer 1

Given that \( k \) divides \( m + 2n \) and \( 3m + 4n \).

Since \( k \) divides \( m + 2n \), it must also divide \( 3(m + 2n) \), which simplifies to \( 3m + 6n \).

We are also given that \( k \) divides \( 3m + 4n \).

If \( k \) divides both \( 3m + 6n \) and \( 3m + 4n \), then \( k \) must divide their difference: \( (3m + 6n) - (3m + 4n) = 2n \). Therefore, \( k \) divides \( 2n \).

Furthermore, since \( k \) divides \( m + 2n \), it must also divide \( 2(m + 2n) \), which is \( 2m + 4n \).

We are given that \( k \) divides \( 3m + 4n \).

Taking the difference between \( 3m + 4n \) and \( 2m + 4n \): \( (3m + 4n) - (2m + 4n) = m \). Therefore, \( k \) divides \( m \).

Consequently, \( m \) and \( 2n \) are both divisible by \( k \).

Correct option: (D) \( m \) and \( 2n \).

For any natural numbers \(m\), \(n\), and \(k\), such that \(k\) divides both \(m + 2n\) and \(3m + 4n\), \(k\) must be a common divisor of

The Correct Option is D

Solution and Explanation

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