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Homework 6 Solution

# Homework 6 Solution - 2.4 Problem 25(a(4 = 2(b(10 = 4(c(13...

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§ 2.4 Problem 25 ( a ): φ (4) = 2. ( b ): φ (10) = 4. ( c ): φ (13) = 12. § 2.4 Problem 26: If n is prime then it is relatively prime to all the numbers 1 , 2 , 3 , . . ., ( n 1), so φ ( n ) = n 1. If n = 1 then φ (1) = 1 negationslash = 1 1. If n is not prime and n > 1 then it has some factor a < n , and gcd( a, n ) = a negationslash = 1. Thus one of the numbers 1 , 2 , 3 , . . ., ( n 1) is not relatively prime to n , and φ ( n ) negationslash = n 1. § 2.4 Problem 27: If p is prime then the only prime factor of p k is p . Thus the only numbers that are not relatively prime to p k are the numbers that have p as a factor. The numbers less than or equal to p k that are not relatively prime to p k are the numbers p, 2 p, 3 p, 4 p, . . ., p k , and there are p k - 1 of these. Thus φ ( p k ) = p k p k - 1 . § 2.4 Problem 30 ( a ): lcm(2 2 · 3 3 · 5 5 , 2 5 · 3 3 · 5 2 ) = 2 5 · 3 3 · 5 5 . ( b ): lcm(2 · 3 · 5 · 7 · 11 · 13 , 2 11 · 3 9 · 11 · 17 14 ) = 2 11 · 3 9 · 5 · 7 · 11 · 13 · 17 14 . ( c ): lcm(17 , 17 17 ) = 17 17 . ( d ): lcm(2 2 · 7 , 5 3 · 13) = 2 2 · 5 3 · 7 · 13. ( e ): lcm(0 , 5) is undefined. (It can’t be 0, because the LCM is defined as a positive integer.) However, some texts would define it to be 0 in this case. ( f ): lcm(2 · 3 · 5 · 7 , 2 · 3 · 5 · 7) = 2 · 3 · 5 · 7. § 2.5 Problem 19: First we need to find the binary representation of 2003.

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Homework 6 Solution - 2.4 Problem 25(a(4 = 2(b(10 = 4(c(13...

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