Intersection of Sets of Integer Multiples/Examples/2 Z cap 7 Z

Examples of Use of Intersection of Sets of Integer Multiples

Let $m \Z$ denote the set of integer multiples of $m$.

Then:

$2 \Z \cap 7 \Z = 14 \Z$

Proof

From Intersection of Sets of Integer Multiples

$2 \Z \cap 7 \Z = \lcm \set {2, 7} \Z$

We have that:

$2 \perp 7$

where $\perp$ denotes coprimality.

Then from LCM equals Product iff Coprime:

$\lcm \set {2, 7} = 2 \times 7 = 14$

Hence the result.

$\blacksquare$

Sources

• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems: Exercise $2$