2. Show that any positive odd integer is of the form 6q+1, or 6q+3, or 69 + 5, where q is
some integer

Question

2. Show that any positive odd integer is of the form 6q+1, or 6q+3, or 69 + 5, where q is
some integer

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Ayla 2 months 2021-08-02T05:08:39+00:00 2 Answers 0 views 0

Answers ( )

    0
    2021-08-02T05:10:11+00:00

    Step-by-step explanation:

    since 6 is a even number when we multiply it with any number we get the answer as a even number and when we add a odd number to a even number we will all ways get odd number.

    Hence proved

    Hope you understand

    0
    2021-08-02T05:10:28+00:00

    Answer:

    According to Euclid’s Division Lemma if we have two positive integers a and b, then there exist unique integers q and r which satisfies the condition a = bq + r where 0 ≤ r < b.

    Let a be the positive odd integer which when divided by 6 gives q as quotient and r as remainder.

    According to Euclid’s division lemma

    a = bq + r

    a = 6q + r………………….(1)

    where, (0 ≤ r < 6)

    So r can be either 0, 1, 2, 3, 4 and 5.

    Case 1:

    If r = 1, then equation (1) becomes

    a = 6q + 1

    The Above equation will be always as an odd integer.

    Case 2:  

    If r = 3, then equation (1) becomes

    a = 6q + 3

    The Above equation will be always as an odd integer.

    Case 3:  

    If r = 5, then equation (1) becomes

    a = 6q + 5

    The above equation will be always as an odd integer.

    ∴ Any odd integer is of the form  6q + 1 or 6q + 3 or 6q + 5.

    Hence proved.

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