Divisibility rule for 22

Question

Divisibility rule for 22: Under what conditions a natural number $N$ is divisible by $22$ ?

My thought is

The divisibility rule for $22$ is that the number is divisible by $2$ and by $11$. Divisibility by $2$ requires that the number ends in $0$, $2$, $4$, $6$ or $8$. Divisibility by $11$ requires that the difference between the sum of the the digits in odd positions and the sum of all the digits in even positions is $0$ or divisible by $11$.

how can i prove that ?

my attempt:

Indeed,

$2 \mid N$ then $N = 2.k$ , and $11\mid N\ $ i.e. $11 \mid 2.k$

or $\textrm{gcd}(2,11) = 1$ then by Lemma gauss $11\mid k$, i.e. $k=11p$ where $N = 2k = 2.11. p = 22p$

Thus

$22\mid N$

Is my reasonable right ?

Answer 1

Yes, that's correct. After applying the divisibility tests for $\,2,11\,$ we know that $\,2,11\mid n,\,$ so it follows that $\,2\cdot 11\mid n,\,$ by Euclid's lemma, or uniqueness of prime factorizations, etc. But one can deduce this much more simply by parity. Namely $\,11\mid n\,\Rightarrow\, n = 11k.\,$ Since further $\,n = 11k\,$ is even, we deduce by parity arithmetic that $\,k\,$ is even $\, k = 2j,\,$ so $\,n = 11k = 11(2j),\ $ thus $\,2\cdot 11\mid n.$

Generally: $\,\ a,b\mid n\iff {\rm lcm}(a,b)\mid n,\ $ the universal property/definition of $\,\rm lcm.$

And $\ \gcd(a,b) = 1\iff {\rm lcm}(a,b) = ab,\ $ which is a special case of $\ \gcd(a,b)\,{\rm lcm}(a,b) = ab.$

Answer 2

The divisibility rule for 22 is that the number is divisible by 2 and by 11. Divisibility by 2 requires that the number ends in 0, 2, 4, 6 or 8. Divisibility by 11 requires that the difference between the sum of the the digits in odd positions and the sum of all the digits in even positions is 0 or divisible by 11.