Question regarding quotient-remainder theorem.

Question

My textbook defines the quotient remainder theorem as follows -

Given any integer *n* and positive integer *d*, there exist unique integers *q* and *r* such that

n = dq + r

where r is a nonnegative integer less than d.

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Here's a question -

For n = 70 and d = 9, **Find integers** ***q*** **and** ***r*** **such that n = dq + r where** r is a nonnegative integer less than d.

According to the book, the answer to this question is q = 7 and r = 7. But q and r cannot have the same value because the quotient remainder theorem specifies that q and r have unique values (which means that they cannot have the same values). Is the textbook's answer incorrect? Or am I misunderstanding the quotient-remainder theorem?

Answer 1

the "unique" here means if you have another pair of (q',r') such that n=dq'+r' then q=q' and r=r'. it doesn't mean q doesn't equal r.

Answer 2

"Unique values" means that there is only one possible choice of q, and only one possible choice of r. It doesn't say anything about whether they are equal.

Answer 3

If n=3 and d=2, is there a solution where d and r are not both 1?

It wouldn't be very useful if those were disallowed.