Download Problem Set 8 The getting is

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 ?