Divisibility rules/Rule 3 for 13 proof

The test works implies divisibility.

$(T - K)/2 * 80 = T \implies 39T = 40K \implies 13 | K$ .

Restrictions so $1 \leq R \leq 39$ : $1 \leq N \leq 1001 \implies 6 \leq K \leq 3042 \implies 6 \leq (39T - 39K) \leq 3042 \\ \implies 1 \leq R \leq 39$

The test does NOT work implies LACK of divisibility. (Uses contrapositive proof.)

For our range of N and consequently our range of K, $13 | N \implies 39 | 3N \implies 78 | K \implies 39T = 40K \implies (T - K)/2 * 80 = T$ .

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Divisibility rules/Rule 3 for 13 proof