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.

&#x200B;

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?
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.
"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.
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.