Question

Now that you know that for every pair of positive integer a and b , there exist a unique pair of whole numbers q and r such that a=bq +r , give example of a and b , wherever possible , satisfying (a) r=0:____________________ (b) q=0:___________________ (c) r>b:____________________ (d) if a<b, what can be said about q and r?___________________

Answer 1

Since the question only requires examples, here they are:

For (a) where r = 0,
a = 8
b=4
q = 2

For (b) where q = 0,
a = 8
b = 10
r = 8

For (c) where r > b

a = 8
b = 1
q = 5
r = 3

For (d) where a < b

a = bq + r

Therefore

bq + r < b

r < b - bq

r < b(1 - q)

Now if q is greater than or equal to 1, then r won't be a whole number.

Hence the only possible value of q is 0

Putting this value in the equation,

r < b(1-0)

r < b

Now putting the value of q in the original equation,

a = bq + r

a = b(0) + r

a = r

Hence in the case where a<b, q = 0 and r = a

Answer 2

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