Question

If two positive integer a and b are written as a=x3 y3 and b=xy3 , x and y are prime numbers. Find the hcf of a and b

Answer 1

HI !

a = x³y³

b = xy³

The common variables with least power is taken 

among x³ and x ,   "x" has the least power  which is  1 .
y³ is common between both "a" and "b" and they are of the same power.

Hence, 

HCF(a,b) = xy³

Answer 2

Answer:

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.