Question

if two positive integers a and b are written as a=x3y2 and b=xy3;x,y are prime numbers,then verify lcm (a,b) x hcf (a,b)=ab

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.

Answer 2

Given that a = x²y³ = x × x × x × y x y
b = xy³ = x × y × y × y

HCF of a and b would be  
= (x³, y², xy³) = x × y × y 
HCF = xy²
HCF is the product of the smallest power of each common prime factor involved in the numbers.

LCM = x³y³
LCM is the lowest common factor of each power.

HCF × LCM 
xy² × x³y³
ab = x⁴y⁵
Thus: ab = HCF ₓ LCM