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Assume we are dealing only with integers.

Using only numerals, variables, logical symbols and the mathematical symbols, I need express the following proposition:

$$100 \text{ is a multiple of } 5 $$

I've tried it, and, in my own words I came up with:

If n is a multiple of 100 then 100 is multiple of 5.

Trying to see if this is right.

3 Answers 3

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"$100$ is a multiple of $5$" by definition:

$$\exists\, x\in \mathbb Z \;( \,5x = 100\,)$$

And we can verify that statement is true by showing the existence of such an integer x:

$$5x= 100 \iff x = \frac{100}{5} = 20, $$ noting that the one and only integer satisfying $5x = 100$ is $x=20$.

Note that there is another way with which to express "$100$ is a multiple of $5$" $\iff$ "$5$ is a divisor of $100$." The notation used to express "$5$ divides $100$" is given by $$5\mid 100$$ And the logical definition given above expresses as much.

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    Thank you I understand right now. I also did it another way. If 100 is a multiple of n, then n is a multiple of 5. 100/20=5, 20/5=4. – 2017-02-06
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    Note that the uniqueness of $x$ ($\exists!$) is quite redundant and is rarely included in the definition of divisibility. – 2017-02-06
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    About the very first comment below my answer: In this case we have the proposition: 100 is a multiple of n ($n=5$) as given, and in this case, it also is trivially true that $n = 5$ is a multiple of 5. But suppose we are told that $6$ is a multiple of $n=3$. That is true because $6 = 2(3),$ but $3$ is not a multiple of $2$. So I think you noticed a relation in this particular case, but the question concerns only how to symbolize the statement that $100$ is a multiple of $5$. – 2017-02-06
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We have,

n = 100k

= 5(20k)

So 100 is a multiple of 5.

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    Your right. But how does this expression be written in symbols. I'm confused – 2017-02-06
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    You are writing in your question symbols are allowed. So we can use. – 2017-02-06
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    Got it. Thank you for your help. – 2017-02-06
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    Mine pleasure :-) – 2017-02-06
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I assume you need it in logic:

$\exists x : 100 = 5 * x$