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adelina 88 [10]
3 years ago
10

Prove that one of every three consecutive positive integer is divisible by 3

Mathematics
1 answer:
Inessa05 [86]3 years ago
5 0
Hello : 
all n in N ; n(n+1)(n+2) = 3a    a in  N  or : <span>≡ 0 (mod 3)
1 ) n </span><span>≡ 0 ( mod 3)...(1)

     n+1 </span>≡ 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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