Find a natural number of the form $n=2^{x} 3^{y} 5^{z}$, knowing that half of it has 30 fewer divisors, a third - 35 fewer, and a fifth - 42 fewer divisors than the number itself.

#
According to the formula from problem $\underline{60537}$ a) we need to solve the system $(x+1)(y+1)(z+1)=x(y+1)(z+1)+30=(x+1) y(z$ $+1)+35=(x+1)(y+1) z+42$.

## Answer

$n=2^{6} \cdot 3^{5} \cdot 5^{4}$