# Inverse Element

For the set of integers \(Z\), an arithmetic operation \(\ast\) is defined as \(a \ast b = a+b+2ab\). How many elements of \(Z\) have an inverse element for \(\ast\)? (The **identity element** is a value \(e\) such that for all values \(n\), \( e \ast n = n \) and \( n \ast e = n .\) The **inverse element** of \(n\) is a value \(m\) such that \( n \ast m = e \) and \( m \ast n = e .\))