GCD and LCM with logical symbols I got one question which says;
Write a formula corresponding to:
1. “a is the greatest common divisor of b and c”
2. “a is the least common multiple of b and c”
Use logical symbols, brackets, · and =. Any other symbols need to be defined.
it seems easy but can someone help me with that ?
 A: Do it in steps:


*

*"$a$ is a divisor of $b$": $\exists k_1\;a\cdot k_1 = b$.

*"$a$ is a divisor of $c$": $\exists k_2\;a\cdot k_2 = c$.

*"$a$ is a common divisor of $b$ and $c$": $(\exists k_1\;a\cdot k_1 = b) \land (\exists k_2\;a\cdot k_2 = c)$.

*"$a$ is the greatest integer satisfying $P(a)$": $P(a) \land (\forall x\;P(x) \implies x \le a)$.


Substituting the third step into the fourth, we get
$$(\exists k_1\;a\cdot k_1 = b) \land (\exists k_2\;a\cdot k_2 = c) \land (\forall x\;((\exists k_1\;x\cdot k_1 = b) \land (\exists k_2\;x\cdot k_2 = c)) \implies x \le a)$$
which represents "$a$ is the greatest common divisor of $b$ and $c$".
There are variations on each of these steps, variations in the notation, and you might be working with a different set of primitives allowed, so you should adjust this to your own needs. For example, if you're not allowed $\le$, then you should proceed from step 3 to trying to write a statement like 


*"$a$ is a common divisor of $b$ and $c$, and if $x$ is a common divisor of $b$ and $c$, then $x$ divides $a$" 


which is an equivalent characterization of GCD.
Writing up LCM is more or less the same, except you stand on your head while you do it.
A: I don't know what language you are talking about, but here is a little something I wrote up out of boredom.  Hope it helps!
Let $a|b$ mean "$a$ divides $b$", let $\geq$ mean "greater than or equal to.
$a |b \wedge a|c \wedge \forall y (y |b \wedge y|c \rightarrow a\geq y))$
"$a$ divides $b$, $a$ divides $c$, and for all $y$, if $y$ divides $b$ and $y$ divides $c$, then $a$ is greater than or equal to $y$"
And for the other one... let $xy=z$ mean $x$ times $y$ equals $z$ and $\leq$ mean less than or equal to.
$\exists  n \exists m (bn=a \wedge cm=a) \wedge \forall y \exists x \exists z (bx=y \wedge cz=y \rightarrow a \leq y) $
"there is an $n$ and an $m$ such that $bn=a$ and $cm=a$ (in other words, $a$ is a multiple of $b$ and $c$), and for all $y$, if $y$ is a multiple of $b$ and $c$ then $a$ is less than or equal to $y$."
