Stupid Divisibility Tricks Marc Renault Shippensburg University MathFest August 2006

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…?

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

Trick #2: Add (Blocks of) Digits Rule of 3: 8362 = 8× × × ≡ (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)

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

d (mod d) 31, -2 75, , d (mod d) ,

d (mod d) 31, -2 74, , , , , d (mod d)

Trick #4: Trim from the Left Test for divisibility by 34: is divisible by = 10 6 × ≡ 10 4 (-2)× (mod 34) 100 ≡ -2 (mod 34)  Trim off leftmost digit  Multiply by 2  Move in 2 places  Subtract

d100 (mod d) d100 (mod d)

duse 62 × × × × × × × × × × × × × × × × × × × 20 duse 622 × × × × 3 × × × × × × × × × × × × × × × × 47 Trick #5: Apply Smaller Divisors Those divisors from 2 to 100 that haven’t been covered by other tricks: