UPSKILL MATH PLUS

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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 $a×b\in \mathrm{ℤ}$ Associativity $a×\left(b×c\right)=\left(a×b\right)×c$ Identity $a×1=a=1×a$ Commutativity $a×b=b×a$ Distributivity $\begin{array}{l}a\left(b+c\right)=a×b+a×c\\ a\left(b-c\right)=a×b-a×c\end{array}$
Example:
1. Suppose we have to simplify the following expression.

$\begin{array}{l}-75×54×4\\ \\ =-75×\left(54×4\right)\phantom{\rule{0.147em}{0ex}}\end{array}$

Applying the commutative property, $a×b=b×a$

$=-75×\left(4×54\right)$

Now applying associative property, $a×\left(b×c\right)=\left(a×b\right)×c$

$\begin{array}{l}=\phantom{\rule{0.147em}{0ex}}\left(-75×4\right)×54\\ \\ =-300×54\\ \\ =-16200\end{array}$

2. Consider another product. $$78 × 19$$.

We can write the number $$19$$ as $$20 - 1$$ to do the multiplication easier.

$\begin{array}{l}=78×19\\ \\ =78×\left(20-1\right)\end{array}$

Applying the distributive property, $a\left(b-c\right)=a×b-a×c$

$\begin{array}{l}=\left(78×20\right)-\left(78×1\right)\\ \\ =1560-78\\ \\ =1482\end{array}$

3. Let us calculate the following product.

$67×\left(-7\right)+\left(-67\right)×3$

Applying distributive property, $a\left(b+c\right)=a×b+a×c$

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