MAT Exam 2023 Paper

Concept:For two numbers in the ratio $6:7$, the HCF is the common multiplier of the two parts, since $6$ and $7$ are coprime. Once the multiplier is found, the actual numbers can be determined and then their LCM can be calculated.Explanation:Let the two numbers be $6x$ and $7x$.Because $6$ and $7$ have no common factor other than $1$, the HCF of $6x$ and $7x$ is $x$.It is given that HCF $= 4$, so $x = 4$.Therefore, the numbers are:$6 \times 4 = 24$ and $7 \times 4 = 28$.Now find the LCM of $24$ and $28$.Prime factorisation gives:$24 = 2^3 \times 3$$28 = 2^2 \times 7$For LCM, take the highest power of each prime factor:$2^3$, $3$, and $7$.Hence, LCM $= 2^3 \times 3 \times 7 = 168$.Answer:The correct option is A. 168.