The ISBN of the fifth edition of Elementary Numbers Theory and Its Applications is 0-32-123Q072, where Q is a?

The ISBN of the fifth edition of Elementary Numbers Theory and Its Applications is 0-32-123Q072, where Q is a digit. Find the value of Q.|||Actually, you can work it out by algebra. The last digit of a 10-digit ISBN - in this case, the '2' at the end - is a check digit. It can always be calculated from the other nine digits. If the complete ISBN is represented by "qrstuvwxyz", where each letter stands for one digit (the hyphens don't count), then





10p + 9r + 8s + 7t + 6u + 5v + 4w + 3x + 2y + z





is always evenly divisible by 11. Inserting the numbers in your example, where p = 0, r = 3, s = 2, and so on (w = Q in particular), you get





10*0 + 9*3 + 8*2 + 7*1 + 6*2 + 5*3 + 4*Q + 3*0 + 2*7 + 2


= 0 + 27 + 16 + 7 + 12 + 15 + 4Q + 0 + 14 + 2


= 4Q + 93





For that to be divisible by 11, there must be an integer n such that 4Q + 93 = 11n. That's a Diophantine equation. Since the numbers are small, you can solve it pretty quickly by trial and error. Try Q = 0, 1, 2, 3, ... until you reach a multiple of 11. It works out that with Q = 7, 4Q + 93 = 121 = 11*11. So the answer is 7.





If the numbers were too large for trial and error, there is a general technique for solving linear Diophantine equations in two variables. It involves the Euclidean algorithm. But that would take another few paragraphs to explain, so I'll stop with that.|||To find "Q" involves googling for the ISBN number. You can't work it out by some formula or other. It happens to be a 7.