Question 1030289
The mystery number would have the form {{{2^p[1]*3^p[2]*5^p[3]}}}.  The number of factors, or divisors, is given by {{{(p[1]+1)(p[2]+1)(p[3]+1)}}}.

==>  {{{(p[1]+1)(p[2]+1)(p[3]+1) = 18}}}.

But 18 = 2*3*3, hence there are {{{(3!)/(1!2!) = 3}}} possibilities, namely

 {{{p[1]=1}}}, {{{p[2]=2}}}, {{{p[3]=2}}};
 {{{p[1]=2}}}, {{{p[2]=1}}}, {{{p[3]=2}}};
 {{{p[1]=2}}}, {{{p[2]=2}}}, {{{p[3]=1}}};

==> The possibilities for the mystery number are 450, 300, or 180.