midterm-solutions

# Let n p 1 p r q 1 q s where none of the p i equal any

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Let n = p 1 . . . p r = q 1 . . . q s , where none of the p i equal any of the q j and p 1 is the smallest among all the primes. Then using the division algorithm to divide q 1 by p 1 , we can find a smaller number k which also has two prime factorizations, which is a contradiction. This works because we know that the remainder in this division is less than p 1 ; in number systems where remainders don’t work this way we don’t have unique factorization. 8. We have seen that an infinite decimal corresponds to a rational number if and only if it is eventually repeating. If instead we consider real numbers expressed in some other base, is this still true? Explain why or why not. Solution. On the one hand, if we have an infinite “decimal” in another base, we can sum a geometric series to find the rational number it represents. On the other hand, if we have two integers and we divide them to find the “decimal” expansion, there are only a finite number of remainders which are possible, so the remainders must eventually repeat, and the expansion is eventually repeating. The answer is yes. 9. Recall the RSA algorithm for cryptography, here reproduced as it appeared on Home- work 3: 1. Choose two distinct primes p and q . (These are usually of similar size, and large; in our examples they will be small.) 2. Compute n = pq ; this will be the modulus. 3. Compute φ ( n ) = ( p - 1)( q - 1), the totient of n . This is the number of numbers among 1 , 2 , . . . , n - 1 which have no factor in common with n . 4. Choose an exponent e such that 1 < e < φ ( n ), where e and φ ( n ) are relatively prime. e is the encoding exponent . 5. Find d such that de 1 (mod φ ( n )), by either trial and error or the Euclidean algorithm. d is called the decoding exponent . Then the public key (which is used to encode messages to be sent to you) is the pair of numbers ( n, e ); the private key is the single number d . To encrypt the message W , compute C = W e (mod n ), and send that. To decode the ciphertext C , compute C d (mod n ).

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