$E \times E = E\Rightarrow E \div E = E$ , right?

Number Theory Level 1

$\begin{aligned} \text{ Even } \times \text{ Even } &=& \text{ Even } \\ \text{ Even } \div \text{ Even } &=& \text{ Even }? \end{aligned}$

It is true that the product of two even numbers is always an even number,
but is it also that the ratio of two even numbers is always an even number too?

No, it is not true Yes, it is true

5 solutions

Peter van der Linden
Jan 20, 2017

Counterexample: 6/2=3

One counter example is enough to disprove the most ugliest of mathematical propositions!

Tapas Mazumdar - 4 years, 2 months ago

Munem Shahriar
Jul 8, 2017

First one is true but the second one is false,

For, $Even$ $\times$ $Even$ $=$ $Even$

Suppose $2$ and $4$ are the two even numbers.

$\Rightarrow 2 \times 4 = 6$ is even.

For, $\dfrac{Even}{Even}$ $=$ $Even$

Suppose $20$ and $4$ are the two even numbers.

$\Rightarrow$ $\dfrac{20}{4}$ $=$ $5$ is odd.

Mohammad Khaza
Jun 30, 2017

10/10=1

that's odd number

Deva Craig
Feb 18, 2017

All even numbers have 2 as a factor. Because of that, we can't assume that all even values divided by another even value also leads to an even value. For example, 2 X 3 = 6, which is an even number. Therefore, 6 ÷ 2 = 3. Also, take note of another number life 18. 18 is equal to 2 X 3 X 3, or 6 X 3, or 18. By dividing 18 by 6, we get a 3.

Yes this is correct. Upvoted!

Note that it's very tempting to say that "Because E x E =E, then dividing both sides by E gives E = E ÷ E", which is not always true.

On the other hand, to disprove a universal statement (a statement that goes something like "this property is always satisfied"), then we just need to mention a counterexample.

Bonus: We know that the product of 2 even numbers is always an even number. Likewise, is it true that the product of 2 odd numbers is always an odd number too?

Pi Han Goh - 4 years, 3 months ago

Kenny O.
Feb 3, 2017

2x/2y=x/y, x/y can be odd (21/7=3), Thus, E/E=E Is not always true.

E × E = E ⇒ E ÷ E = E E \times E = E\Rightarrow E \div E = E E × E = E ⇒ E ÷ E = E , right?

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$E \times E = E\Rightarrow E \div E = E$ , right?