Question

Show that any positive odd integer is of the form of (6m+1) or(6m+3) or (6m+5)​

Answer 1

Answer:

Let s be any positive odd integer. On dividing s by 6, let m be the quotient and r be the remainder.

By Euclid’s division lemma,

s = 6m + r, where 0 ≤ r ˂ 6

So we have, s = 6m or s = 6m + 1 or s = 6m + 2 or s = 6m + 3 or s = 6m + 4 or s = 6m + 5.

6m, 6m + 2, 6m + 4 are multiples of 2, but s is an odd integer.

Again, s = 6m + 1 or s = 6m + 3 or s = 6m + 5 are odd values of s

. Thus, any positive odd integer is of the form (6m + 1) or (6m + 3) or (6m + 5), where m is any odd integershow-that-any-positive-odd-integer-is-of-the-form-6m-1-or-6m-3-or-6m-5-where-m-is-some-integer.

Answer 2

ANSWER:

By Euclid's Division Lemma,

a = bq + r where 0< or = r < 6 since b=6

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

Let m = q

If r = 0, a = 6q + 0 (even)

If r = 1, a = 6q + 1 (odd)

If r = 2, a = 6q + 2 (even)

If r = 3, a = 6q + 3 (odd)

If r = 4, a = 6q + 4 (even)

If r = 5, a = 6q + 5 (odd)

Since, 6 is even, any multiple of 6 is also even and when we add an even number to another even number, the result is also even.

Therefore, 6q, 6q+2 and 6q+4 are even.

A number has to be either even or odd so if 6q+1, 6q+3 and 6q+5 are not even, so they are odd.

Hence, proved that any positive odd integer is of the form 6m+1, 6m+3 and 6m+5

Hope it helps!!