One post tagged with "outreach"

Based on some of the problems from Prof. Dragos Ghioca's problem set 1.

Question 1#

Let $a, b \in \mathbb{N}$. Show that if $\gcd(a,b) = \text{lcm}(a,b)$, then $a = b$.

Question Explanation#

$a, b$ here are natural numbers, i.e. the ones that are used to count things: $1, 2, \dotso$

Now, consider the set of all numbers that divide both $a$ and $b$. The maximum of this set is defined to be $\gcd(a,b)$. Formally: $\gcd(a,b) := \max \{ x : x | a\ \text{and}\ x | b \}$.

Similarly, consider the set of all numbers that are divisible by both $a$ and $b$. The minimum of this set is defined to be $\text{lcm}(a,b)$. Formally: $\text{lcm}(a,b) := \min \{ x : a | x\ \text{and}\ b | x \}$.