If 10!=2p⋅3q⋅5r⋅7s, then

Exponent of 2 is102+1022+1023=5+2+1=8Exponent of 3 is103+1032=3+1=4Exponent of 5 is105=2Exponent of 7 is107=1The number of divisors of 10! is (8 + 1) (4 + 1) (2 + 1) (1 + 1) = 270. The number of ways of putting N as a product of two natural numbers is 270/2 = 135.