# What is the difference between these two statements; which is true and which is false?

I am learning about predicates and am somewhat a beginner in proofs. I would like your help on proving why statement 2 is false. These are the two statements:

$\text{statement 1: }(\exists x_1 \in \mathbb{N}, x_1|165) \wedge(\exists x_2 \in \mathbb{N}, 7|x_2)\\ \\ \text{statement 2: }\exists x \in \mathbb{N}, x|165 \wedge 7|x \\ \text{Note: a|b is read as a divides b, and is defined as: } \exists k \in \mathbb{Z}, a.k = b\\ \text{I believe statement 1 is true, since, there exists a natural number that divides 165 (165 itself),}\\ \text{and there exists another number that is divisible by 7 (7 itself). So statement 1 is true.}\\ \text{Statement 2 is false, but I'm not sure why, and how can we prove it to be false?}\\ \text{So I started by trying to prove its negation to be true, i.e:}\\ \forall x \in \mathbb{N}, x \nmid165 \vee 7 \nmid x \\ \Leftrightarrow \forall x\in \mathbb{N} , x|165 \implies 7\nmid x$ Some discussion(using what I've learnt so far about proofs): $\text{Let x be a natural number, and assume x|165 , i.e } \\\exists k1 \in \mathbb{Z},x.k1 = 165 \text{ . Let k1 be this number.} \\ \text{I want to show } 7 \nmid x, \text{i.e } \neg(\exists k_2 \in \mathbb{Z}, 7.k2 = x) \Leftrightarrow (\forall k_2 \in \mathbb{Z}, 7.k_2 \neq x).... \\ \text{Now, how can I use my assumption to reach what I want to show? Thank you for your directions.}$

• Do you have the fundamental theorem of arithmetic? – Eric Towers Apr 13 '18 at 23:26