Question

If two of the integers a and b are written as a=x3y2 and b=xy3; x, y are prime numbers, then find the H.C.F (a, b)

Answer 1

When getting LCM using indices, we get the highest indices of the two and when getting HCF we get the lowest power of the two unknowns.
a = x³y² , b= xy³
LCM
Comparing indices of x and y in numbers a and b. 
a : x's index is 3 whereas y's index is 2
b: x's index is 1 and y's index is 3.
Comparing the two: the highest index of x is 3 and the highest index of y is 3.
LCM = The highest indices of the unknowns (x and y) 
LCM =x³y³
HCF = The lowest indices of x and y
The lowest index of x is 1 and the lowest index of y is 2. 
HCF = xy²
ab = x³y²(xy³) = x⁴y⁵
HCF × LCM = x³y³(xy²) = x⁴y⁵
Thus ab = LCM × HCF