Question

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Answer 1

6q+1:6 is divisible by 2 but 1 is not divisible by 2 so it is a odd number�

6q+2:6 is divisible by 2 and 2 is also divisible by 2 so it is a even number

6q+3:6 is divisible by 2 but 3 is not divisible by 2 so it is a odd number

6q+4:6 is divisible by 2 and 4 is also divisible by 2 it is a even number

6q+5:6 is divisible by 2 but 5 is not divisible by 2 so it is a odd number

So odd numbers will in form of 6q + 1, or 6q + 3, or 6q + 5

Answer 2

Using Euclid's Division Algorithm,

a = bq + r , where a is the positive integer.

Let b = 6

Therefore, r = 0,1,2,3,4,5 [ Since, 0 ≤ r < b ]

When r = 0

==> a = 6q

==> a = 2(3q)

==> a = 2m, where m = 3q { even }

When r = 1

==> a = 6q + 1 { odd }

When r = 2

==> a = 6q + 2

==> a = 2(3q + 1)

==> a = 2m, where m = 3q + 1 { even }

When r = 3

==> a = 6q + 3 { odd }

When r = 4

==> a = 6q + 4

==> a = 2(3q + 2)

==> a = 2m, where m = 3q + 2 { even }

When r = 5

==> a = 6q + 5 { odd }

Therefore, any positive integer is of the form (6q + 1), (6q + 3), (6q + 5).