Question

show that one and only one out of N N + 2 or n + 4 is divisible by 3 Where n is any positive integer​

Answer 1

Hey

mate

answer is given below

Solution: We know that any positive integer is of the form 3q or 3q + 1 or 3q + 2 for some integer q & one and only one of these possibilities can occur

Case I : When n = 3q

In this case, we have,

n=3q, which is divisible by 3

n=3q

= adding 2 on both sides

n + 2 = 3q + 2

n + 2 leaves a remainder 2 when divided by 3

Therefore, n + 2 is not divisible by 3

n = 3q

n + 4 = 3q + 4 = 3(q + 1) + 1

n + 4 leaves a remainder 1 when divided by 3

n + 4 is not divisible by 3

Thus, n is divisible by 3 but n + 2 and n + 4 are not divisible by 3

Case II : When n = 3q + 1

In this case, we have

n = 3q +1

n leaves a reaminder 1 when divided by 3

n is not divisible by 3

n = 3q + 1

n + 2 = (3q + 1) + 2 = 3(q + 1)

n + 2 is divisible by 3

n = 3q + 1

n + 4 = 3q + 1 + 4 = 3q + 5 = 3(q + 1) + 2

n + 4 leaves a remainder 2 when divided by 3

n + 4 is not divisible by 3

Thus, n + 2 is divisible by 3 but n and n + 4 are not divisible by 3

Case III : When n = 3q + 2

In this case, we have

n = 3q + 2

n leaves remainder 2 when divided by 3

n is not divisible by 3

n = 3q + 2

n + 2 = 3q + 2 + 2 = 3(q + 1) + 1

n + 2 leaves remainder 1 when divided by 3

n + 2 is not divsible by 3

n = 3q + 2

n + 4 = 3q + 2 + 4 = 3(q + 2)

n + 4 is divisible by 3

Thus, n + 4 is divisble by 3 but n and n + 2 are not divisible by 3

Answer 2

Step-by-step explanation:

Euclid's division Lemma any natural number can be written as: .

where r = 0, 1, 2,. and q is the quotient.

thus any number is in the form of 3q , 3q+1 or 3q+2.

case I: if n =3q

n = 3q = 3(q) is divisible by 3,

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

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

case II: if n =3q + 1

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

n + 2 = 3q + 1 + 2 = 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 III: if n = 3q + 2

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 3.

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

......

Hence, it is solved

THANKS