Stupid Divisibility Tricks Marc Renault Shippensburg University Math

Stupid Divisibility Tricks Marc Renault Shippensburg University Math. Fest August 2006

Rule of 3 Rule of 7 Rule of 19 161 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…?

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 Examine last 6 digits

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 9) 10 ≡ -1 (mod 11) 100 ≡ 1 (mod 33) 100 ≡ 1 (mod 99) 100 ≡ -1 (mod 101) 1000 ≡ -1 (mod 7) 1000 ≡ -1 (mod 13) 1000 ≡ 1 (mod 27) 1000 ≡ 1 (mod 37) 1000 ≡ -1 (mod 77) 1000 ≡ -1 (mod 91) Add digits Add pairs of digits Add triples of digits

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 To test divisibility by d find an inverse of 10 (mod d).

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

d 3 7 9 11 13 17 19 21 23 27 29 31 33 37 39 41 43 47 49 51 100 -1 (mod d) 1, -2 4, -3 1 1 3, -10 8, -9 4 4 3, -20 10 40, -3 8 d 53 57 59 61 63 67 69 71 73 77 79 81 83 87 89 91 93 97 99 101 100 -1 (mod d) -9 4 -2 -20 -10 -20 -8 -10 40 1 -1

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

d 7 13 14 19 21 32 33 34 35 48 100 (mod d) 2 -4 2 5 -5 4 1 -2 -5 4 d 49 51 52 53 95 96 97 98 99 101 100 (mod d) 2 -2 -4 -6 5 4 3 2 1 -1

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