Divisibility by 19

Theorem

Let $n$ be an integer expressed in the form:

$n = 100 a + b$

Then $n$ is divisible by $19$ if and only if $a + 4 b$ is divisible by $19$.

Let $a, b, c \in \Z$.

$\ds 100 a + b$

$=$

$\ds 19 c$

$\ds \leadstoandfrom \ \ $

$\ds 400 a + 4 b$

$=$

$\ds 19 \paren {4 c}$

Multiply by $4$

$\ds \leadstoandfrom \ \ $

$\ds 399 a + a + 4 b$

$=$

$\ds 19 \paren {4 c}$

Separate the $a$ values

$\ds \leadstoandfrom \ \ $

$\ds 19 \paren {21 a} + a + 4 b$

$=$

$\ds 19 \paren {4 c}$

Factor out $19$

$\ds \leadstoandfrom \ \ $

$\ds a + 4 b$

$=$

$\ds 19 \paren {4 c - 21 a}$

Subtract

$\blacksquare$