Study Guide · Chapter 20
2.18 Divisibility by 19
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Rule (Osculator, additive): Multiply last digit by 2, add to remaining number. Repeat. Example: Check 1,007. Last digit 7, ×2=14. Remaining=100. 100+14=114. 114 ÷ 19 = 6 → divisible by 19.
Example 2: Check whether 1,045 is divisible by 19. Last digit 5, ×2=10. Remaining=104. 104+10=114. Again: last digit 4, ×2=8. Remaining=11. 11+8=19, which is divisible by 19 (in fact equal to it) → 1,045 is divisible by 19 (1045=19×55 ✓).
Example 3: Check whether 1,083 is divisible by 19. Last digit 3, ×2=6. Remaining=108. 108+6=114, and we already know from Example 2 that 114 is divisible by 19 → 1,083 is divisible by 19 (1083=19×57 ✓).
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