Do the multiplication in easier way — lesson. Mathematics State Board, Class 7.

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Integers have five remarkable properties on multiplication operation. With the help of those properties, we can find the product value easily.

Let us see how to apply those properties to do the multiplication more easily.

Now let \(a\) and \(b\) are integers. Their multiplicative properties are as follows:

Integer property	Over multiplication
Closure	$a \times b \in ℤ$
Associativity	$a \times (b \times c) = (a \times b) \times c$
Identity	$a \times 1 = a = 1 \times a$
Commutativity	$a \times b = b \times a$
Distributivity	$\begin{array}{l} a (b + c) = a \times b + a \times c \\ a (b - c) = a \times b - a \times c \end{array}$

Example:

1. Suppose we have to simplify the following expression.

\begin{array}{l} - 75 \times 54 \times 4 \\ = - 75 \times (54 \times 4) \end{array}

Applying the commutative property,

a \times b = b \times a

= - 75 \times (4 \times 54)

Now applying associative property,

a \times (b \times c) = (a \times b) \times c

\begin{array}{l} = (- 75 \times 4) \times 54 \\ = - 300 \times 54 \\ = - 16200 \end{array}

2. Consider another product. \(78 × 19\).

We can write the number \(19\) as \(20 - 1\) to do the multiplication easier.

\begin{array}{l} = 78 \times 19 \\ = 78 \times (20 - 1) \end{array}

Applying the distributive property,

a (b - c) = a \times b - a \times c

\begin{array}{l} = (78 \times 20) - (78 \times 1) \\ = 1560 - 78 \\ = 1482 \end{array}

3. Let us calculate the following product.

67 \times (- 7) + (- 67) \times 3

Applying distributive property,

a (b + c) = a \times b + a \times c

\begin{array}{l} = 67 ((- 7) + (- 3)) \\ = 67 (- 7 - 3) \\ = 67 (- 10) \\ = - 670 \end{array}

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