The valid declarations in this question are int $x;, int _123;, and int central_sales_region_Summer_2005_gross_sales;. The first declaration, int $x;, is valid because it starts with a valid character, the dollar sign. The second declaration, int _123;, is valid because it starts with a valid character, the underscore. The last declaration, int central_sales_region_Summer_2005_gross_sales;, is valid because it starts with a valid character, a letter. The other declarations, int 123; and int #dim;, are not valid because they start with a number and a pound sign, respectively, which are not allowed in variable names.