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

✓ Step 1 :

Given number is 6q+1, 6q+ 3and 6q + 5.

Let a be a positive integer.

Let b = 6

Apply Euclid's divisor algorithm on a and 6.

a = 6q + r....(1)

where r is reminder and r = 0, 1, 2, 3, 4, 5

✓ Step 2 :

Plugging in (1) and checking whether it is an odd number

• a = 6q+0

= 6q

= 2 × 3q

Since 2 is a factor of 6q. So, it is an even

number.

✓ Step 3:

Plugging r = 1 in(1) and checking whether it is an odd number

•a = 6q+1

Since 2 is not a factor of 6q+1. So it is an

odd number.

✓ Step 4:

Plugging r = 2 in (1) and checking whether it is an odd number

•a = 6q + 2

= 2 × (3q + 1)

Since 2 is a factor of 6q + 2. So, it is an even

number.

✓ Step 5:

Plugging r = 3 in(1) and checking whether it is an odd number

•a = 6q+4

= 2 × (3q + 2)

Since 2 is a factor of 6q+4. So, it is an even

✓ Step 6:

Plugging r = 4 in (1) and checking whether it is an odd number

•a =6q+4

= 2 × (3q + 2)

Since 2 is a factor of6q+4. So, it is an even

number

✓ Step 7:

Plugging r = 5 in(1) and checking whether it is an odd number

•a = 6q+5

Since 2 is not a factor of 6q+5. So it is an

odd number.

Final Answer-

Therefore, any odd integer can be expressed in the form, 6q+1 or 6q+ 3or 6q+5

Answer 2

Answer:

Any odd number can be shown in this form.

Step-by-step explanation:

Because any number cannot be divided by 2 when expressed in this form.