What numbers x and y make the equation y~2 +y=x~3~x~2 true? Renaissance mathematicians approached these equations by introducing clock or modular arithmetic. On a conventional clock with 12 hours, we know that 9 o'clock plus 4 hours is 1 o'clock rather than 13 o'clock. We write this as 9+4= 1 modulo 12. Consider a clock with 1 hours labelled 0,1,2,3,4,5 and 6. The question now is how many pairs of numbers (x,y) chosen from the possible hours on this clock will make the equation y~2+y=x~3 - x~2 true. For example, if we take y=3, then 3~2+3 = 9+3 = 12. On the 7-hour clock, this comes out at 5 o'clock. But if we put x=6 in the other side of the equation then 6~3-6~2=216-36= 180, which also has remainder 5 on division by 7. So we say that the pair(x,y)=(6,3) is a solution of the equation y~2+y=x~3-x~2 modulo 7.
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