Question

How to prove the uniqueness of q and r in a=bq+r?

Answer 1

Here's your answer

To prove: The uniqueness of q and r in Euclid's Division Lemma.

Proof: To prove the uniqueness of q and r, let us assume that there is another pair q1 and r1 of non negative integers satisfying the same relation i.e.,

a=bq+r, 0≤r<b

We shall now prove that r1=r and q1=q

We have,

a=bq+r and a=bq1+r1

bq+r=bq1+r1

r1-r=bq-bq1

r1-r=b(q-q1)

r1/b-r

r1-r=0[Because 0≤r<b and 0≤r1<b. this gives 0≤r1-r<b]

r1=r

Now, r1=r

-r1=-r [Multiplying both sides by (-1)]

a-r1=a-r [Adding a on both sides]

bq1=bq

q1=q

Hence, the representation a=bq+r, 0≤r<bis unique.

In the proof I can't understand the underlined steps. Please help!

Thanks in advance.

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