hw1 - 3-176 4 = 242 3-(1870-242 7) 4 = 242 31-1870 4 r = 31...

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MATH 3360 HOMEWORK SOLUTION 1 (Warning: You need to specify the domain of the variables in your answer (e.g. p35, E5). It is especially necessary if you write something like x = - 4 + 9 8 n .) p11, E1 Base case: 4! = 24 > 16 = 2 4 . Inductive step: suppose n ! > 2 n , then ( n + 1)! = ( n + 1) × n ! > ( n + 1) × 2 n > 2 n +1 . p11, E2 Base case: 1 3 = 1 = ( 1 × 2 2 ) 2 . Inductive step: suppose the formula holds for n , then in case of n + 1: 1 3 + 2 3 + ... + ( n + 1) 3 =(1 3 + 2 3 + ... + n 3 ) + ( n + 1) 3 = 1 4 n 2 ( n + 1) 2 + ( n + 1) 3 =( n + 1) 2 [ 1 4 n 2 + n + 1] =( n + 1) 2 ( 1 2 n + 1) 2 = 1 4 ( n + 1) 2 ( n + 2) 2 p29, E6(ii) 43137 = 21063 × 2 + 1011 21063 = 1011 × 20 + 843 1011 = 843 × 1 + 168 843 = 168 × 5 + 3 168 = 3 × 56 So their gcd is 3. p33, E4(ii) 1870 = 242 × 7 + 176 242 = 176 × 1 + 66 176 = 66 × 2 + 44 66 = 44 × 1 + 22 44 = 22 × 2 1
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d = 22 and 22 = 66 - 44 = 66 - (176 - 66 × 2) = 66 × 3 - 176 = (242 - 176) × 3 - 176 = 242
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Unformatted text preview: 3-176 4 = 242 3-(1870-242 7) 4 = 242 31-1870 4 r = 31 ,s =-4 other solutions are possible but they have to be integers. p35, E5 (i) x = 9 n,y =-8 n , n Z (ii) x =-4 + 9 n,y = 6-8 n , n Z p35, E6(i) x = 5 ,y =-4 p35, E7(i) The gcd of 267 and 112 is 1. x =-39 + 112 n,y = 93-276 n , n Z p35, E8 The gcd of 203 and 119 is 7. So only (iii) has a solution. x = 2 ,y =-3. p36, E20 First of all, since ar + bs is divisible by d , it cannot be any positive number less than d . It is possible to nd r,s s.t. ar + bs = d by the algorithm we learned in this section. 2...
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This note was uploaded on 09/05/2011 for the course MATH 3360 taught by Professor Billera during the Spring '08 term at Cornell University (Engineering School).

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hw1 - 3-176 4 = 242 3-(1870-242 7) 4 = 242 31-1870 4 r = 31...

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