Question

1. Show that square of any integer if of the form 3m or 3m+1.

2. Show that one and only one out of n , n+2,n+4 is divisible by 3.

Answer 1

Answer: 1. is in the images and 2. is below!

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Step-by-step explanation:

Let n be any positive integer and b=3

n =3q+r

where q is the quotient and r is the remainder

0_ <r<3

So the remainders may be 0,1 and 2

So n may be in the form of 3q, 3q=1,3q+2

CASE-1

IF N=3q

n+4=3q+4

n+2=3q+2

Here n is only divisible by 3

CASE 2

if n = 3q+1

n+4=3q+5

n+2=3q=3

Here only n+2 is divisible by 3

CASE 3

IF n=3q+2

n+2=3q+4

n+4=3q+2+4

=3q+6

Here , only n+4 is divisible by 3.

HENCE IT IS JUSTIFIED THAT ONE AND ONLY ONE AMONG n,n+2,n+4 IS DIVISIBLE BY 3 IN EACH CASE.