(a) Consider the ring $R$ of polynomials in $n$ variables with integer coefficients. Prove that the polynomial $f \left( x _ { 1 } , x _ { 2 } , \ldots , x _ { n } \right) = x _ { 1 } x _ { 2 } \cdots x _ { n }$ has $2 ^ { n + 1 } - 2$ non-constant polynomials in $R$ dividing it. (b) Let $p _ { 1 } , p _ { 2 } , \ldots , p _ { n }$ be distinct prime numbers. Then show that the number $N = p _ { 1 } p _ { 2 } ^ { 2 } p _ { 3 } ^ { 3 } \cdots p _ { n } ^ { n }$ has $( n + 1 ) !$ positive divisors.
(a) Consider the ring $R$ of polynomials in $n$ variables with integer coefficients. Prove that the polynomial $f \left( x _ { 1 } , x _ { 2 } , \ldots , x _ { n } \right) = x _ { 1 } x _ { 2 } \cdots x _ { n }$ has $2 ^ { n + 1 } - 2$ non-constant polynomials in $R$ dividing it.\\
(b) Let $p _ { 1 } , p _ { 2 } , \ldots , p _ { n }$ be distinct prime numbers. Then show that the number $N = p _ { 1 } p _ { 2 } ^ { 2 } p _ { 3 } ^ { 3 } \cdots p _ { n } ^ { n }$ has $( n + 1 ) !$ positive divisors.