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| quote: | Originally posted by -=M=-
how'd that work? |
elementary number theory
n1 - initial number
n2 - number formed by the same digits but in different oder
fact 1:
|n1-n2| ≡ 0 (mod 9)
fact 2:
∀k∈N, 9 | k ↔ 9 | (d1+d2+d3+d4+...+dn) where d1,d2,d3...dn are digits of k
in this case
k = |n1-n2|, therefore 9 | k, therefore 9 | (d1+d2+...+dn)
step v asks you to remove one digit. without loss of generality suppose it's d2.
then d2 = 9p - (d1+d3+d4+...+dn), where p∈N
fact 3:
∀d∈DIGIT, 0 ≤ d ≤ 9
in this case 0 ≤ 9p - (d1+d3+d4+...+dn) ≤ 9
it can be easily shown that d2 is uniquely defined by the above double inequality.
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The Amazing Phone Number Trick
