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When dealing with integer multiplication, the process is straightforward – you are simply multiplying numbers together. However, things get tricky when you introduce negative numbers into the mix. The key rule to remember is that multiplying two numbers with the same sign (both positive or both negative) will give you a positive result, while multiplying numbers with different signs will give you a negative result.

For example, in the exercise \( -2(5)(-3) \), you start by multiplying \( -2 \) and \( 5 \). Despite \( 5 \) being positive, the product of it and a negative number \( -2 \), according to the rule, will be negative: \( -2 \times 5 = -10 \). This is the first essential step in integer multiplication.

Understanding the properties of negative numbers is crucial in arithmetic operations such as multiplication. A fundamental property is that the product of two negative numbers is positive. This might initially seem counter-intuitive but makes sense when you consider that a negative number represents the opposite of a positive number.

So, if we consider \( -2 \) to be the opposite of \( 2 \) and \( -3 \) to be the opposite of \( 3 \) as in our exercise, multiplying these negatives is the same as changing the direction twice, which lands you back in the positive: \( -10 \times -3 = 30 \). Remember, this property holds true regardless of how many negative numbers are being multiplied, as long as the count is even, the product will remain positive.

Arithmetic operations, such as addition, subtraction, multiplication, and division, are the building blocks of mathematics. In these operations, multiplication has a unique feature: it is repetitive addition. For instance, \( 5 \times 3 \) means adding the number \( 5 \) three times.

However, when negative numbers are involved, you should consider both the rules of integer multiplication and the properties of negative numbers. In our exercise, after understanding that \( -2 \times 5 = -10 \) follows the rule of different signs leading to a negative product, it’s applied again to \( -10 \times -3 \). This step echoes reinforcing the concept of negative property – multiplying two negatives results in a positive. It’s the consistency of these rules across all arithmetic operations that allows us to predict and understand the outcomes of complex calculations.