By F. Goodman

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7. Modular Arithmetic We are all familiar with the arithmetic appropriate to the hours of a clock: If it is now 9 o’clock, then in 7 hours it will be 4 o’clock. Thus in clock arithmetic, 9 + 7 = 4. The clock number 12 is the identity for clock addition: Whatever time the clock now shows, in 12 hours it will show the same time. Multiplication of hours is not quite so familiar an operation, but it does make sense: If it is now 12 o’clock, then after 5 periods of seven hours it will be 11 o’clock, so 5 7 D 11 in clock arithmetic.

Fix a natural number n > 1. Think of a clock face with n hours (labeled 0; 1; 2; : : : ; n 1) and of circumference n. Imagine taking a number line, with the integer points marked, and wrapping it around the circumference of the clock, with 0 on the number line coinciding with 0 on the clock face. Then the numbers : : : ; 3n; 2n; n; 0; n; 2n; 3n; : : : on the number line all overlay 0 on the clock face. The numbers : : : ; 3n C 1; 2n C 1; n C 1; 1; n C 1; 2n C 1; 3n C 1; : : : on the number line all overlay 1 on the clock face.

Q 0 q/d D 0, so q 0 q D 0. ■ We have shown the existence of a prime factorization of any natural number, but we have not shown that the prime factorization is unique. This is a more subtle issue, which is addressed in the following discussion. The key idea is that the greatest common divisor of two integers can be computed without knowing their prime factorizations. 8. A natural number ˛ is the greatest common divisor of nonzero integers m and n if (a) ˛ divides m and n and (b) whenever ˇ 2 N divides m and n, then ˇ also divides ˛.