Show that one and only one out of n, n+2 or n +4 is divisible by 3, where n is any positive integer. een​

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.
een​

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Autumn 7 months 2021-10-13T17:03:52+00:00 1 Answer 0 views 0

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    2021-10-13T17:05:08+00:00

    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.

    if this helps you

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