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find! In fact, these are all the roots of any sort Remember: A polynomial equation can never have more roots than its degree That includes not only the rational roots, but the irrational and complex roots

133 Give an ef cient algorithm to compute the least common multiple of two n-bit numbers x and y, that is, the smallest number divisible by both x and y What is the running time of your algorithm as a function of n 134 On page 35, we claimed that since about a 1/n fraction of n-bit numbers are prime, on average it is suf cient to draw O(n) random n-bit numbers before hitting a prime We now justify this rigorously Suppose a particular coin has a probability p of coming up heads How many times must you toss it, on average, before it comes up heads (Hint: Method 1: start by showing that the correct expression is i(1 p)i 1 p Method 2: if E is the average number of coin tosses, show that i=1 E = 1 + (1 p)E) (N 1)! 1 (mod N ) (a) If p is prime, then we know every number 1 x < p is invertible modulo p Which of these numbers are their own inverse (b) By pairing up multiplicative inverses, show that (p 1)! 1 (mod p) for prime p (c) Show that if N is not prime, then (N 1)! 1 (mod N ) (Hint: Consider d = gcd(N, (N 1)!))

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Problem 16-24 Suppose a parallel RX circuit (shown by the generic block diagram of Fig 16-12) has R = 300 and X = 200 The ac supply voltage is 500 V rms What is the total current drawn from the ac supply First, find the square of the absolute-value impedance, remembering the formula for parallel circuits: Z 2 = R 2X 2/(R 2 + X 2) = 360,000/1300 = 277 The absolute-value impedance Z is the square root of 277, or 166 The total current is therefore I = E/Z = 50/166 = 301 A rms Problem 16-25 What are the rms currents through the resistance and the reactance, respectively, in the circuit described in Problem 16-24 The Ohm s Law formulas for dc will work here For the resistance, IR = E/R = 500/300 = 167 A rms For the reactance, IX = E/X = 500/( 200) = 25 A rms Note that these currents don t add up to 301 A, the total current The reason for this is the same as the reason ac voltages don t add arithmetically in ac circuits that contain reactance The constituent currents, IR and IX, differ in phase Vectorially, they add up to 301 A rms, but arithmetically, they don t

This is an open-book quiz You may (and should) refer to the text as you solve these problems Don t hurry! You ll find worked-out answers in App C The solutions in the appendix may not represent the only way a problem can be figured out If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it! 1 Rewrite each of the following equations in binomial to the nth form (a) (x 2 + 6x + 9)2 = 0 (b) (x 2 4x + 4)3 = 0 (c) (16x 2 24x + 9)4 = 0 2 What are the real roots for each of the equations stated in Prob 1 What is the multiplicity in each case 3 Rewrite each of the following equations in binomial factor form (a) (x 2 3x + 2)2 = 0 (b) ( 3x 2 5x + 2)5 = 0 (c) (4x 2 9)3 = 0 4 What are the real roots for each of the equations stated in Prob 3 What is the multiplicity in each case 5 State the real roots of the following equation Also state the real solution set X and the multiplicity of each root What is the degree of the equation (x 3/2)2(2x 7)2(7x)3( 3x + 5)5 = 0 6 State the real roots of the following equation Also state the real solution set X and the multiplicity of each root What is the degree of the equation (x + 4)(2x 8)2(x /3 + 12)3 = 0

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