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Stupid Divisibility Tricks Marc Renault Shippensburg University MathFest August 2006

Stupid Divisibility Tricks Marc Renault Shippensburg University MathFest August 2006

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Page 1: Stupid Divisibility Tricks Marc Renault Shippensburg University MathFest August 2006

Stupid Divisibility Tricks

Marc Renault

Shippensburg University

MathFest

August 2006

Page 2: Stupid Divisibility Tricks Marc Renault Shippensburg University MathFest August 2006

Rule of 3Rule of 7 161Rule of 19

Other numbers?Other categories of tricks?

L.E. Dickson 1919 History of the Theory of Numbers

Martin Gardner 1962Scientific American 2 – 12

Internet, number theory texts, liberal studies texts

Useful…?

Page 3: Stupid Divisibility Tricks Marc Renault Shippensburg University MathFest August 2006

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

Page 4: Stupid Divisibility Tricks Marc Renault Shippensburg University MathFest August 2006

Trick #2: Add (Blocks of) DigitsRule 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 digits10 ≡ -1 (mod 11)

100 ≡ 1 (mod 11)100 ≡ 1 (mod 33) Add pairs of digits100 ≡ 1 (mod 99)100 ≡ -1 (mod 101)

1000 ≡ -1 (mod 7)1000 ≡ -1 (mod 13)1000 ≡ 1 (mod 27) Add triples of digits1000 ≡ 1 (mod 37)1000 ≡ -1 (mod 77)1000 ≡ -1 (mod 91)

Page 5: Stupid Divisibility Tricks Marc Renault Shippensburg University MathFest August 2006

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).

Page 6: Stupid Divisibility Tricks Marc Renault Shippensburg University MathFest August 2006

d 10-1 (mod d)3 1, -27 5, -29 111 -113 4, -917 -519 221 -223 727 -829 331 -333 1037 -1139 441 -443 -304749 551 -5

d 10-1 (mod d)5357 4059 661 -66367 -2069 771 -7737779 881 -883 258789 9, -8091 -9939799 10101 -10

Page 7: Stupid Divisibility Tricks Marc Renault Shippensburg University MathFest August 2006

d 100-1 (mod d)3 1, -27 4, -39 111 113 3, -1017 8, -919 421 423 3, -2027 1029 -20313337 10394143 40, -347 84951

d 100-1 (mod d)53 -957 459616367 -269 -20717377 -1079818387 -2089 -891 -1093 409799 1 101 -1

Page 8: Stupid Divisibility Tricks Marc Renault Shippensburg University MathFest August 2006

Trick #4: Trim from the Left

Test for divisibility by 34:587044- 10 77044 - 14 5644 - 10 544 - 10 34587044 is divisible by 34

587044 = 106×5 + 87044 ≡ 104(-2)×5 + 87044 (mod 34)

100 ≡ -2 (mod 34)

Trim off leftmost digitMultiply by 2Move in 2 placesSubtract

Page 9: Stupid Divisibility Tricks Marc Renault Shippensburg University MathFest August 2006

d 100 (mod d)7 213 -414 219 521 -532 433 134 -235 -548 4

d 100 (mod d)49 251 -252 -453 -695 596 497 398 299 1101 -1

Page 10: Stupid Divisibility Tricks Marc Renault Shippensburg University MathFest August 2006

d use6 2 × 312 3 × 418 2 × 922 2 × 1124 3 × 826 2 × 1328 4 × 730 3 × 1036 4 × 938 2 × 1942 2 × 2144 4 × 1145 5 × 946 2 × 2354 2 × 2755 5 × 1156 7 × 858 2 × 2960 3 × 20

d use62 2 × 3163 7 × 965 5 × 1366 2 × 3 × 1168 4 × 1770 7 × 1072 8 × 974 2 × 3775 3 × 2576 4 × 1978 2 × 3982 2 × 4184 4 × 2185 5 × 1786 2 × 4388 8 × 1190 9 × 1092 4 × 2394 2 × 47

Trick #5: Apply Smaller Divisors

Those divisors from 2 to 100 that haven’t been covered by other tricks: