Question

Show that any positive odd integers is of the form 8q+1or 8q+3or8q+7.where q Is some integers

Answer 1

Correct Question :

Show that any positive odd integer is of the form 8q + 1 or 8q + 3 or 8q + 5 or 8q + 7.

Solution :

Let ' a ' be any positive integer.

Here, b = 8 ( from question ).

BY USING EUCLID'S DIVISION LEMMA :

→ a = bq + r, where 0 ≤ r < d.

Possible values of r = 0, 1, 2, 3, 4, 5, 6 and 7.

Now,

Case : 1

a = 8q + 0 _______ [ EVEN ]

Case : 2

a = 8q + 1 ______ [ ODD ]

Case : 3

a = 8q + 2 _______ [ EVEN ]

Case : 4

a = 8q + 3 _______ [ ODD ]

Case : 5

a = 8q + 4 _______ [ EVEN ]

Case : 6

a = 8q + 5 __________ [ ODD ]

Case : 7

a = 8q + 6 ____ [ EVEN ]

Case : 8

a = 8q + 7 ________ [ ODD ]

From above, it's clear that any positive odd integer is of the form 8q + 1, 8q + 3, 8q + 5 or 8q + 7.

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