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Unformatted text preview: s . Exercise 3. Show that the equation x 1 + x 2 + + x j = n has n-1 j-1 solutions in positive integers. Note: solutions dier if they dier in at least one index, i.e. (1 , 1 , 2) diers from (2 , 1 , 1). Exercise 4. Show that there are 2 n-1 ways of writing n as a sum of strictly positive integers where order of summands is taken into account (i.e., 1 + 2 + 1 is dierent from 2 + 1 + 1 and both are dierent from 1 + 1 + 2). Exercise 5. Prove that the number of positive fractions 1 that have denominator n (when reduced to lowest terms) is (1) + (2) + (3) + + ( n ) . Exercise 6. Assume m ( m ) = n ( n ). Show m = n . Exercise 7. Let N be a positive integer and let f and F be arithmetic functions. Show that the following are equivalent: 1. F ( n ) = b N/n c k =1 f ( kn ) for all n . 2. f ( n ) = b N/n c k =1 ( k ) F ( nk ) for all n . (Here is the Mobius function.)...
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- Spring '08