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Stupid Divisibility Tricks

Stupid Divisibility Tricks. Marc Renault Shippensburg University. MathFest August 2006. Rule of 3 Rule of 7 161 Rule of 19 Other numbers? Other categories of tricks? L.E. Dickson 1919 History of the Theory of Numbers Martin Gardner 1962 Scientific American 2 – 12

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Stupid Divisibility Tricks

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  1. Stupid Divisibility Tricks Marc Renault Shippensburg University MathFest August 2006

  2. Rule of 3 Rule of 7 161 Rule of 19 Other numbers? Other categories of tricks? L.E. Dickson 1919 History of the Theory of Numbers Martin Gardner 1962 Scientific American 2 – 12 Internet, number theory texts, liberal studies texts Useful…?

  3. Trick #1: Examine Ending Digits 2, 5, 10 divide 10 Examine last digit 4, 20, 25, 100 divide 100 Examine last 2 digits 8, 40 divide 1000 Examine last 3 digits 16, 80 divide 10,000 Examine last 4 digits 32 divides 100,000 Examine last 5 digits 64 divides 1,000,000 Examine last 6 digits

  4. Trick #2: Add (Blocks of) Digits Rule of 3: 8362 = 8×1000 + 3×100 + 6×10 + 2 ≡ 8 + 3 + 6 + 2 (mod 3) 10 ≡ 1 (mod 3) 10 ≡ 1 (mod 9) Add digits 10 ≡ -1 (mod 11) 100 ≡ 1 (mod 11) 100 ≡ 1 (mod 33) Add pairs of digits 100 ≡ 1 (mod 99) 100 ≡ -1 (mod 101) 1000 ≡ -1 (mod 7) 1000 ≡ -1 (mod 13) 1000 ≡ 1 (mod 27) Add triples of digits 1000 ≡ 1 (mod 37) 1000 ≡ -1 (mod 77) 1000 ≡ -1 (mod 91)

  5. Trick #3: Trim from the Right Test for divisibility by 7: 6034 - 8 595 -10 49 6034 = 10×603 + 4 mod 7… 10×603 + 4 ≡ 0  (-2)10×603 + (-2)4 ≡ 0  603 + (-2)4 ≡ 0 To test divisibility by d find an inverse of 10 (mod d).

  6. d10-1 (mod d) 3 1, -2 7 5, -2 9 1 11 -1 13 4, -9 17 -5 19 2 21 -2 23 7 27 -8 29 3 31 -3 33 10 37 -11 39 4 41 -4 43 -30 47 49 5 51 -5 d10-1 (mod d)53 57 40 59 6 61 -6 63 67 -20 69 7 71 -7 73 77 79 8 81 -8 83 25 87 89 9, -80 91 -9 93 97 99 10 101 -10

  7. d100-1 (mod d) 3 1, -2 7 4, -3 9 1 11 1 13 3, -10 17 8, -9 19 4 21 4 23 3, -20 27 10 29 -20 31 33 37 10 39 41 43 40, -3 47 8 49 51 d100-1 (mod d) 53 -9 57 4 59 61 63 67 -2 69 -20 71 73 77 -10 79 81 83 87 -20 89 -8 91 -10 93 40 97 99 1 101 -1

  8. Trick #4: Trim from the Left Test for divisibility by 34: 587044 - 10 77044 - 14 5644 - 10 544 - 10 34 587044 is divisible by 34 587044 = 106×5 + 87044 ≡ 104(-2)×5 + 87044 (mod 34) 100 ≡ -2 (mod 34) • Trim off leftmost digit • Multiply by 2 • Move in 2 places • Subtract

  9. d100 (mod d) 7 2 13 -4 14 2 19 5 21 -5 32 4 33 1 34 -2 35 -5 48 4 d100 (mod d) 49 2 51 -2 52 -4 53 -6 95 5 96 4 97 3 98 2 99 1 101 -1

  10. Trick #5: Apply Smaller Divisors Those divisors from 2 to 100 that haven’t been covered by other tricks: duse 62 2 × 31 63 7 × 9 65 5 × 13 66 2 × 3 × 11 68 4 × 17 70 7 × 10 72 8 × 9 74 2 × 37 75 3 × 25 76 4 × 19 78 2 × 39 82 2 × 41 84 4 × 21 85 5 × 17 86 2 × 43 88 8 × 11 90 9 × 10 92 4 × 23 94 2 × 47 duse 6 2 × 3 12 3 × 4 18 2 × 9 22 2 × 11 24 3 × 8 26 2 × 13 28 4 × 7 30 3 × 10 36 4 × 9 38 2 × 19 42 2 × 21 44 4 × 11 45 5 × 9 46 2 × 23 54 2 × 27 55 5 × 11 56 7 × 8 58 2 × 29 60 3 × 20

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