1. Solve the following diophantine equations, if possible. This means: either say that the equation has no solutions, or describe all solutions (x, y) where x and y ar both integers. (a) 24x+32y=16 (b) 64x-8y = 3 (c) 8x-7y=4 (d) 64x+8y = 4 2. A woman has $4.85 in dimes and quarters. (a) Set up and solve the appropriate diophantine equation for the number of dimes and number of quarters y that she could possibly be carrying. (b) Some of the solutions you described in the last part have (x, y) where x could possibly be negative or y could possibly be negative. These solutions obviously do not make sense in the context of the problem. Write down inequalities involving the the variable t in your solution that guarantee that both x and y are greater than or equal to 0. (c) What is the maximum number of coins that she could be carrying? The minimum number? 3. We know from last class that if a is composite then 2-1 is composite. This is accomplished by remembering the (very important!) formula x" - 1 +2-1 Use this formula to produce three different divisors of the number 230 1 1073741823 For each divisor, explain what your ris, and what your n is. Calculate the quotient also. 1