Divisibility by 19
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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 $19$.
Proof
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 |
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $19$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $19$