homework8 - 5). 6. Let f be a multiplicative function. Show...

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THEORY OF NUMBERS, Math 115 A Homework 8 NOT Due Friday December 07 1. Find the sums and numbers of divisors of 196, 2 100 and 10!. 2. Show that the number of divisors of n is odd if and only if n is a square. Characterize also the numbers whose sum of divisors is odd. 3. Let σ k ( n ) denote the sum of k -th powers of divisors of n . For example, σ 2 (12) = 1 2 + 2 2 + 3 2 + 4 2 + 6 2 + 12 2 = 210 . (a) Show that σ k is multiplicative. (b) Show that if p is a prime then σ k ( p r ) = p kr + k - 1 p k - 1 . (c) Derive the general formula for σ k ( n ) in terms of the prime factorization of n . Use it to recalculate σ 2 (12), and find σ 3 (12). 4. Find all integers n between 1 and 100 with μ ( n ) = 1. 5. For how many consecutive integers can μ be equal to zero? For how many can it be different from zero? (Hint: one of these questions is related to a problem in homework
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Unformatted text preview: 5). 6. Let f be a multiplicative function. Show that X d | n ( d ) f ( d ) = Y p | n , p prime (1-f ( p )) . 7. Find the primes p and q if n = pq = 4 , 386 , 607 and ( n ) = 4 , 382 , 136. 8. Suppose that in an RSA encoded message one of the encoded blocks C turns out to be not relatively prime to n = pq . (a) Show that using that C (and the encyphering exponent and modulus e and n ) a cryptanalist can factor n , hence break the code. (b) If p and q have, say, 100 digits, what is the probability that such a bad block will actually occur? 1...
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