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Problem Set 8 The secret ofgetting ahead is getting started. The secret ofgetting start is breaking your complex overwhelming tasks into small manageable tasks, and then starting on the first one. --Mark Twain Prove or Disprove and Salvage if Possible. J. If mp < np and p is positive then m < n. True in N? In Z? 2. If a=' b (mod n) and c =' d (mod n) then a + c =' b + d (mod n). 3. If a=' b (mod n) and c =' d (mod n) then ac =' bd (mod n). 4 . ......,(alb) and ......,(alc) => => 81(n2 - I). 5 . ......,(2In) ......,(albc). Ingenuity. 6. Given x we can compute x 48 by computing x' x' x ... X· x. This method involves 47 mUltiplications. We can do better, using onJy 24 multiplications: First use 23 multiplications to compute x24 = X·X "'X, then multiply once more to find x 48 = x24 ·x 24 What is the smallest number of multiplication operations needed to compute x 48 ?