Mathematics
lhmrocks
2016-09-24 15:28:36
Prove that one of every three consecutive positive integer is divisible by 3
ANSWERS
pamadov
2016-09-24 17:58:03

hello :  all n in N ; n(n+1)(n+2) = 3a    a in  N  or : ≡ 0 (mod 3) 1 ) n ≡ 0 ( mod 3)...(1)      n+1 ≡ 1 ( mod 3)...(2)       n+2 ≡ 2 ( mod 3)...(3) by (1), (2), (3)  : n(n+1)(n+2) ≡ 0×1×2   ( mod 3)   : ≡ 0 (mod 3) 2) n ≡ 1 ( mod 3)...(1)      n+1 ≡ 2 ( mod 3)...(2)       n+2 ≡ 3 ( mod 3)...(3) by (1), (2), (3)  : n(n+1)(n+2) ≡ 1×2 × 3  ( mod 3)   : ≡ 0 (mod 3) , 6≡ 0 (mod)  3) n  ≡ 2 ( mod 3)...(1)      n+1 ≡ 3 ( mod 3)...(2)       n+2 ≡ 4 ( mod 3)...(3) by (1), (2), (3)  : n(n+1)(n+2) ≡ 2×3 × 4  ( mod 3)   : ≡ 0 (mod 3) , 24≡ 0      (mod3)

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