1.

Show that one and only one out of n, n+1 or n+2 is divisible by 3 where n is any positive integer .

Answer»

we know that any +ve integer is of the form 3q,3q+1,3q+2 

case 1.

when n=3q {in this case n is divisible by 3,} 

n+1=3q+1 {n+1 is not divisible by three} 

n+2=3q+2 {n+2 is not divisible by 3} 

thus n is divisible by 3 but n+1,n+2 is not divisible by 3 

case-2.

when n =3q+1 {n is not divisible by 3} 

n+1=3q+1+1=3q+2 {n+1 is not divisible by three} 

n+2=3q+1+2=3q+3 

n+2=3(q=1) {n+2 is divisible by 3} 

thus n+2 is divisible by 3,but n and n+1 is not divisible by 3 

case-3

when n=3q+2 {n is not divisible by 3} 

n+1=3q+2+1=3q+3 

n+1=3(q+1) {n+1 is divisible by three} 

n+2=3q+2+2=3q+4 

n+2=3(q+1)+1 {n+2 is not divisible by 3} 

thus,n+1 is divisible by 3 but n and n+2 is not divisible by3


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