2023/005) What is the difference of numbers if the HCF of two numbers is 9 and their LCM is 270 if the sum of the numbers is 99?

HCF is 9 so the 2 numbers are 9x, 9y where HCF(x,y) = 1 and x and y >0 and without loss of generality let x > y

product of the 2 numbers are 9xy = 270

or $xy = 30\cdots(1)$

sum of the 2 numbers are 9(x+y) = 99 or $x + y = 11\cdots(2)

we have $(x-y)^2 = (x+y)^2 - 4 * xy = 11^2 - 120 = 1$

so $x - y = 1\cdots(3)$

so we get x = 6 and y = 5 and numbers are $54, 45$