Number theory - Diophantine Equations(SELLING BOOKs)

We actually didn't need equation.

It was clear that n-2 has to be a multiple of 5.

So either 77 or 97 can be the answer.

So now if we take 77 then after giving books to Fred , Verna is left with 60 books.

Now it again is clear that 60-6 has to be a factor of 5 but is is not.

So now 97 is left so it is the answer.

Let B = total books, F = books sold to Fred, and J = books sold to Joan. So we want to solve for B: F = 2 + (B-2)/5 F = (10 + B - 2)/5 = (8 + B)/5 J = 6 + (B-F-6)/5 J = (30 + B - F - 6)/5 J = (24 + B - F ) /5 Substituting from the first into the second: J = (24 + B - (8+B)/5)) / 5 J = (120 + 5B - 8 - B) / 25 25 J = 112 + 4 B 25 J - 4 B = 112

By observation we see that J = 4 and B = -3 solves this, but -3 is not a valid number of books.

Increase J by 4 and B by 25 to get J = 8 and B = 22 Then F = 2 + 20/5 = 6.

But the problem states Fred scored more books, so we go to the next value, J = 12 and B = 47, which gives F = 11, so he's getting closer!

Another try gives J = 16 and B = 72, which gives F = 16, tied with Joan!

So one last try J = 20 and B = 97, giving F = 21, finally beating out that evil witch Joan by 1 book. 97 books in the collection!