The sum of two numbers is $2000$ and thier LCM is $21879.$ Find the numbers.
My attempt: Let two numbers be $x, 2000-x.$ Product of two numbers is equal to the product of their lcm and hcf. So, $x(2000-x)=21879*hcf.$
Now we have two variables and one equation. So I am stuck.
But the book simply considers $x(2000-x)=21879,$ thereby $x=1989,11.$
My question is – Is it implied anywhere in the question that hcf is $1$ or is the question incomplete?
More impotantly, can we have any other pair of numbers whose sum is $2000$ and lcm is $21879 (hcf <> 1)?$
Best Answer
Note that $21879$ and $2000$ are coprime. So this means that $x$ and $2000-x$ are coprime (simply because their gcd must divide $x+(2000-x)=2000$).