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  • Prove that one of any three consecutive positive integers must be divisible by 3.

Prove that one of any three consecutive positive integers must be divisible by 3.

Answers (1)

Let three consecutive integers are n, n + 1, n + 2 when n is a natural number
  i.e., n = 1, 2, 3, 4, ….
for n = 1,         chosen numbers :(1, 2, 3) and 3 is divisible by 3   
for n = 2,         chosen numbers :(2, 3,4) and 3 is divisible by 3    
for n = 3,         chosen numbers :( 3, 4, 5) and 3 is divisible by 3  
for n = 4,         chosen numbers :(4, 5, 6) and 6 is divisible by 3
for n = 5,         (chosen numbers :(5, 6, 7) and 6 is divisible by 3   
Proof:

Case 1 : Let n is divisible by 3, it means n can be written as :
n = 3m,
Now, n +1 = 3m + 1 ; it gives remainder 1, when divided by 3
Now, n +2 = 3m + 2; it gives remainder 2, when divided by 3

Case 2: Let n + 1 is divisible by 3, it means n +1 can be written as :
n +1 = 3m,
Now, n = 3m – 1= 3(m – 1) + 2 ; it gives remainder 2, when divided by 3
Now, n +2 = 3m + 1; it gives remainder 1, when divided by 3

Case 3: Let n + 2 is divisible by 3, it means n +2 can be written as :
n +2 = 3m,
Now, n = 3m – 2= 3(m – 1) + 1 ; it gives remainder 1, when divided by 3
Now, n +1 = 3m – 1=3(m – 1) + 2  ; it gives remainder 2, when divided by 3

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