Question

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

Answer 1

Answer:

Let

a1=6q+0

a2=6q+1

a3=6q+2

a4=6q+3

a5=6q+4

a6=6q+5

here, a1 , a3, and a5 are are divisible by 2

2 is divisible 2,4

2 is not divisible 1, 3,5

6q+1, 6q+3, 6q+5 is odd numbers

Answer 2

All positive integers can be represented in the form of:

p = aq + b where $0\leq b< a$ (Euclid's division Lemma)

Here, p = 2n + 1 (as 2n is even)

a = 6

6q + 0 = 6q = 2(3q)

6q + 1 = 2(3q) + 1 is odd

6q + 2 = 2(3q + 1) which is even

6q + 3 = 6q + 2 + 1 = 2(3q + 1) + 1 which is odd

6q + 4 = 2(3q + 2) whichh is even

6q + 5 = 6q + 4 + 1 = 2(3q + 2) + 1 which is odd

Since all positive integers can be represented in this form, the above accounts for all positive integers

So, all odd numbers can be only 6q + 1, 6q = 3 or 6q + 5. Hope it helps :)

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