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

2. Rule of 3 Rule of 7161 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 10Examine last digit 4, 20, 25, 100 divide 100Examine last 2 digits 8, 40 divide 1000Examine last 3 digits 16, 80 divide 10,000Examine last 4 digits 32 divides 100,000Examine last 5 digits 64 divides 1,000,000Examine 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. d 10 -1 (mod d) 31, -2 75, -2 91 11-1 134, -9 17-5 192 21-2 237 27-8 293 31-3 3310 37-11 394 41-4 43-30 47 495 51-5 d 10 -1 (mod d) 53 5740 596 61-6 63 67-20 697 71-7 73 77 798 81-8 8325 87 899, -80 91-9 93 97 9910 101 -10

7. d 100 -1 (mod d) 31, -2 74, -3 91 111 133, -10 178, -9 194 214 233, -20 2710 29-20 31 33 3710 39 41 4340, -3 478 49 51 d 100 -1 (mod d) 53-9 574 59 61 63 67-2 69-20 71 73 77-10 79 81 83 87-20 89-8 91-10 9340 97 991 101 -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 = 10 6 ×5 + 87044 ≡ 10 4 (-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) 72 13-4 142 195 21-5 324 331 34-2 35-5 484 d100 (mod d) 492 51-2 52-4 53-6 955 964 973 982 991 101 -1

10. duse 62 × 3 123 × 4 182 × 9 222 × 11 243 × 8 262 × 13 284 × 7 303 × 10 364 × 9 382 × 19 422 × 21 444 × 11 455 × 9 462 × 23 542 × 27 555 × 11 567 × 8 582 × 29 603 × 20 duse 622 × 31 637 × 9 655 × 13 662 × 3 × 11 684 × 17 707 × 10 728 × 9 742 × 37 753 × 25 764 × 19 782 × 39 822 × 41 844 × 21 855 × 17 862 × 43 888 × 11 909 × 10 924 × 23 942 × 47 Trick #5: Apply Smaller Divisors Those divisors from 2 to 100 that haven’t been covered by other tricks: