If x=ab3 and y=a3b, where a and b are prime numbers, then . . .

Question : If x=ab³ and y=a³b, where a and b are prime numbers, then [HCF(x,y)-LCM(x,y)] is equal to :

(a) 1-a³b³

(b) ab(1-ab)

(c) ab-a⁴b⁴

(d) ab(1-ab)(1+ab)

CBSE 2024

Solution : 

x=ab³
y=a³b

HCF = ab
LCM = a³b³

HCF(x,y)-LCM(x,y)
=ab-a³b³
=ab[1-a²b²]
=ab[1²-(ab)²]
=ab[(1-ab)(1+ab)]
=ab(1-ab)(1+ab) 

Hence, (d) ab(1-ab)(1+ab), would be the correct option.