My teacher gave me a question at which I am stuck:
The sum of two positive integers is 52 and their LCM is 168, then what
are the numbers?
I studied a concept that the product of LCM and HCF is equal to the product of the numbers, but I am not able to apply that over here.
Can someone help me with this?
Thanks for the help.
Best Answer
We see that $168 = 2^3 \cdot 3^1 \cdot 7^1$, so we must find two numbers, each of whose prime factorizations has at most three $2$s, at most one $3$, and at most one $7$. Further, at least one of the prime factorizations has to have $2^3$, at least one has to have $3^1$, and at least one has to have $7^1$. Luckily this narrows down the search space considerably.
We eventually notice that $28 = 2^2 \cdot 7^1$ and $24 = 2^3 \cdot 3^1$, and $28 + 24 = 52$.