Question

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show that any +ve odd integer is of the form 8q+1 or 8q+3 or 8q+5 or 8q+7 where q is some integer.

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Answer 1

Let q be 1
8q+1
=8(1)+1
=8+1
=9 which is +ve odd integer
let q be 7
8q+3
=8(7)+3
=56+3
=59 which is also +ve odd integer
Let the resultant odd +ve integer be 101
8q+5=101
8q=96
q=12
Hence proved

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Answer 2

Step-by-step explanation:

a = bq+r where, 0 <= r < b

so all possible values of a if b = 8

a = 8q+0, 8q+1, 8q+2, 8q+3, 8q+4, 8q+5, 8q+6, 8q+7

as it is asked for all odd values:

8q+0, 8q+2, 8q+4, 8q+6 will not be included as all these are even values

so answer will be:

a = 8q+1, 8q+3, 8q+5, 8q+7