MAT135-Assignment9-Solutions

MAT135-Assignment9-Solutions - MATH 135, Fall 2010 Solution...

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MATH 135, Fall 2010 Solution of Assignment #9 Problem 1 . Suppose that p and q are prime numbers with p > q . Suppose also that n = pq and φ ( n ) = ( p - 1)( q - 1). (a) Prove that p + q = n - φ ( n ) + 1 and p - q = p ( p + q ) 2 - 4 n . (b) If n = 1 281 783 203 and φ ( n ) = 1 281 711 600, determine p and q . Solution. (a) In the first equation, RS = n - φ ( n ) + 1 = pq - ( p - 1)( q - 1) + 1 = pq - ( pq - p - q + 1) + 1 = p + q = LS In the second equation, RS = p ( p + q ) 2 - 4 n = p p 2 + 2 pq + q 2 - 4 pq = p p 2 - 2 pq + q 2 = p ( p - q ) 2 = p - q = LS since p > q . (b) From (a), p + q = n - φ ( n ) + 1 = 1 281 783 203 - 1 281 711 600 + 1 = 65 604 and p - q p ( p + q ) 2 - 4 n = p 65 604 2 - 4(1 281 783 203) = 4 = 2 Thus, 2 p = ( p + q ) + ( p - q ) = 65 606 whence p = 32 803. Lastly, q = ( p + q ) - p = 98 096 - 74 623 = 32 801. Problem 2 . Use Fermat’s Little Theorem and the Square and Multiply Algorithm to show that the integer 3599 is not prime, without testing each prime p 3599 to see if is a factor. (Hint: compute 2
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MAT135-Assignment9-Solutions - MATH 135, Fall 2010 Solution...

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