Question

show that any positive odd integer is of the form 8q+1,8q+3,8q+5,8q+7 for some integer q

Answer 1

a=bq+r
0<or=r<b.
b=8
0,or=r<8.
r=0,1,2,3,4,5,6,7.
1.a=8q+0=8q.
2.a=8q+1
3.a=8q+2
4.a=8q+3
5.a=8q+4
6.a=8q+5
7.a=8q+6
8.a=8q+7.

From the above cases 8q,8q+2,8q+4,8q+6 are divisible by 2 and hence are even integers.

But 8q+1,8q+3,8q+5 and 8q+7 are odd integers and are not divisible by 2.

Hence,we can say that any positive odd integers are of the form 8q+1,8q+3,8q+5,8q+7 for some integer q.

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Answer 2

Answer:

Step-by-step explanation:

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According to division algorithm,

we can write any number ‘a’ in the form

Step-by-step explanation:

a = 8q + r

where q is any integer and 0 <= r <= 7. So r can be 0, 1, 2, 3, 4, 5, 6 or 7.

Thus, a can be written as

a = 8q

a = 8q+2

a = 8q+3

a = 8q+4

a = 8q+5

a = 8q+6

a = 8q+7

So any odd integer can be written as any one of the remaining forms which are (8q+1, 8q+3, 8q+5 or 8q+7.)