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By Garett P.

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Example: n = 10 = 2 1 . 51 and m = 25 = 52. Thus (10,25) = 5. Two numbers n and m that have no common factors are called relatively prime, mutually prime or coprime. In this case the GCD equals 1. Example: (6,35) = (2·3, 5'7) = 1. 3) for the LCM picks the exponent fi, and vice versa. 5) '= nm . ·5) = 2; [4,10] = [~2, 2'~] = 20; 2· 20 = 4 . 10. Check! 6) which is easily verified. 6). The same is true for all primes occurring in nmk. 6) then follows from the fundamental theorem of arithmetic. 7) which is proved by the same reasoning.

From Basel who worked for most of his life in St. Petersburg in Russia. There was a time when the present author was much impressed by the fact that the sum of the reciprocal divisors of P is always 2: 1 ""l; - = 2! 8) (Here, for once, the exclamation mark does not do any harm because 2! 7), because in a sum over all divisors d of a given number n, the divisor d may be replaced by n/d. 9) 2:fCn/d). 10) 1+2+3+6=6+3+2+1. 8). Check: 1 + ~ + ~ + ~ = 2. Check! It is remarkable that the sum of the reciprocal divisors of a perfect number always equals 2, no matter how large it iso This implies that perfect numbers cannot have too many small divisors, as we already know.

This can happen only if n and m have a greatest common divisor greater than 1. This brings us to our next topic. 20 2. 4 The Greatest Common Divisor Another important relation between integers is their greatest common divisor (GCD). 4) i because for the GCD to divide both n and m it cannot have the factor Pi more often than it is contained in either n or m, whichever is less. Example: n = 10 = 2 1 . 51 and m = 25 = 52. Thus (10,25) = 5. Two numbers n and m that have no common factors are called relatively prime, mutually prime or coprime.

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Automorphic Representations and L-functions by Garett P.


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