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I am not at all sure that there are any rules that apply to integers in isolation. Any rules that exist are in the context of binary operations like addition or multiplication of integers.
The various operations within the expression are carried out using the order of operations: BIDMAS (UK) or PEMDAS (US).
Parenthesis, Exponents, Multiplication or Division(whichever come first from left to right*), Addition or Subtraction *
You need the rules of multiplication as well as of addition. But multiplication of integers can be viewed as repeated addition. Thus, if p/q and r/s are two rational numbers then their sum is(p*s + q*r)/(q*s)
There are different rules for different operations.