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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…?
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
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)
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).
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
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
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
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
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: