Question

Number of pairs of positive integers $(p,q)$ whose LCM (Least common multiple) is $8100$, is “$K$”. Then number of ways of expressing $K$ as a product of two co-prime numbers is ...............

• A. 2
• B. 6
• C. 4
• D. 8

Answer 1

Answer: (A)

Given,
$L.C.M(p,q)=8100=2^2{3}^4{.5}^{2}Let,P=2^{a_1}{3}^{b_1}{.5}^{c_1},q=2^{a_2}{3}^{b_2}{5}^{c_2}\Rightarrow max\{a_1,a_2\}=2\Rightarrow 5ways\left[\right(2,0),(2,1\left)\right(2,2),(0,2),(1,2\left)\right]\Rightarrow max\{b_1,b_2\}=4\Rightarrow 9ways\left[\right(4,0),(4,1),(4,2),(4,3),(4,4),(0,4),(1,4),(2,4),(3,4\left)\right]\Rightarrow max\{c_1,c_2\}=2\Rightarrow 5ways\left[\right(2,0),(2,1),(2,2),0,2),(1,2)]\therefore K=3^2{.5}^2=225$ ways