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MT2_sln

# MT2_sln - Math 55 Practice Midterm Solutions(prepared by...

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Math 55 Practice Midterm Solutions (prepared by Nick Meyer) 1a. FALSE. There is a one-to-one correspondence between the half-open interval H = [0 , 1) R and the circle C R 2 ; this correspondence can be demonstrated by the bijective function f : H C that sets f ( x ) = (cos 2 πx, sin 2 πx ) for x H. Since H is a nonempty interval of the real numbers, it is uncountable; since C is in one-to-one correspondence with H , it must also be. 1b. TRUE. The two statements are contrapositives of each other, hence logically equivalent. 1c. FALSE. Let p = 23; there is no prime q strictly between p and p + 6 = 29. (Note: there is nothing special about 6 here. For any positive n , we have that 2 , 3 , 4 , . . . , n all divide n !; hence 2 | ( n ! + 2) , 3 | ( n ! + 3) , . . . , n | ( n ! + n ). We thus have constructed a sequence of n - 1 consecutive composite numbers – namely, n ! + 2 through n ! + n. ) 1d. TRUE. 30 = 2 × 3 × 5; 77 = 7 × 11. We must merely find an m which does not have any of 2 , 3 , 5 , 7 , 11 as a factor. 13 naturally suggests itself. 1e. FALSE. Indeed, everyone knows how Bob encrypted his message. To encrypt message M , Bob calculates ciphertext C = M e mod n and sends C to Alice. n and e

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