# 74-hw2s - MATH 74 HOMEWORK 2: DUE FEBRUARY 5 1(a). Student...

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MATH 74 HOMEWORK 2: DUE FEBRUARY 5 1(a). Student A gets 3/10, student B gets 10/10. Student A unwittingly changed the problem (to something simpler). Student B did not. [Something this short was all I wanted.] Aside. To emphasize Student A’s mistake, it helps to notice that his “reformula- tion” is true with any number in place of 5 , but the original statement is not— for example, 12 | 6 2 but 12 6 | 6 . Student A has eliminated the diﬃculty of the problem by replacing it with a simpler true statement. Aside. A statement of the form “if X, then Y” is called an implication. Student A proved “if Y, then X,” which is called the converse of the ﬁrst implication. It is in general a completely diﬀerent statement. Student B proved “if Y is false, then X is false” which is called the contrapositive of the original implication. It is equivalent to the original implication. See Solow for more examples. 1(b). We prove: if 5 6 | a , then 5 6 | a 2 . Proof. By grade school division there are q N 0 and 0 r < 5 such that a = 5 q + r. As 5 6 | a , the number r is not zero. Thus r is one of 1 , 2 , 3 , 4. We consider each of these cases in turn.

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## This homework help was uploaded on 04/02/2008 for the course MATH 74 taught by Professor Courtney during the Fall '07 term at University of California, Berkeley.

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74-hw2s - MATH 74 HOMEWORK 2: DUE FEBRUARY 5 1(a). Student...

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