CBSE Class 10 Answered
Dear Student,
According to
Given positive integers a and b, there exist unique integers q and r
satisfying a = bq + r,
This lemma is true for any two integers
Now, we want to show that any even integer is of the form '2q' and
any odd integer is of the form '2q + 1'
So we take the two integers a and 2
When we divide a with 2, the possible values of remainder are 0 and 1, which means
a = 2q + 0
Or
a = 2q + 1
Also, 2q is always an even integer for any integral value of q and so 2q + 1 will always be an odd integer. (Note that 'even integer + 1' is always an odd integer)
Hence, when a is a positive even integer it will always be of the form 2q
And when a is positive odd integer it will always be of the form 2q + 1
Regards,
Team
Topper Learning