If the L.C.M. of (p,q) is (r^{2}t^{4}s^{2}), where (r,s,t) are prime numbers and (p,q) are the positive integers then the number of ordered pair ( (p,q) ) is: Permutations and Combinations (Mathematics)

(p, q)=r^2 t^4 s^2 \\

For L.C.M. of

p, q

to contain

r^2

the possible powers of

r

p

and

q

are

(0,2),(2,0),(1,2),(2,1),(2,2)

is 5

\\

Similarly the possible powers of

t:(0,4),(4,0),(1,4),(4,1)

(2,4),(4,2),(3,4),(4,3),(4,4)

is 9

\\

Similarly the possible powers of

s

(like

r

) is 5

\\

Required number of orderd pairs

=5 \times 9 \times 5=225

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