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

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Let a be any positive integer and b = 6.

Then,

by Euclid’s algorithm,

a = 6q + r,

for some integer q ≥ 0,

and r = 0, 1, 2, 3, 4, 5,

because 0≤r<6.

Now substituting the value of r,

we get,

If r = 0, then a = 6q

Similarly, for r= 1, 2, 3, 4 and 5,

the value of a is 6q+1, 6q+2, 6q+3, 6q+4 and 6q+5, respectively.

If a = 6q, 6q+2, 6q+4, then a is an even number and divisible by 2.

A positive integer can be either even or odd.

Therefore,

Any positive odd integer is of the form of 6q+1, 6q+3 and 6q+5, where q is some integer.

Answer 2

Answer:

Let a be the positive odd integer which when divided by 6 gives q as quotient and r as remainder. but here, a=6q+1 & a=6q+3 & a=6q+5 are odd.

Step-by-step explanation: