1.

Show that one only one of n,n+2,n+4 is divisible by 3​

Answer»

Let n be any integer. let it divide by 3

by using Euclid's DIVISION lemma :

n = 3q + R

0_< r<3

r=0,1,2

n=3q or 3q+1or 3q+2

Case1-:

let n=3q

here n = 3q is DIVISIBLE by 3

n+2=3q+2 is not divisible by 3

n+4= 3q+4 = 3q+3+1= 3(q+1)+1 is not divisible by 3.

CASE 2-:

let n = 3q+ 1

here n = 3q+ 1 is not divisible by 3

n+2 = 3q+1+3= 3q+3 = 3(q+1) is divisible by 3

n+4= 3q+1+4= 3q +5= 3(q+1)+2 is not divisible by 3

Case 3-:

let n =3q+2

here n= 3q+2 is not divisible by 3

n+2 =3q+2+2= 3q+4= 3(q+1)+1 is not divisible by 3

n+4= 3q+2+4 =3q+6= 3(q+2) is divisible by 2

Hence, it is CLEAR that one and only one out of n,n+2,n+4 is divisible by 3


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