Unformatted text preview: ﬁnitely many primes, say, p 1 ,p 2 ,...,p n . Form the number N = p 2 p 3 p 4 ··· p n + p 1 p 3 p 4 ··· p n + p 1 p 2 p 4 ··· p n + ··· + p 1 p 2 p 3 ··· p n1 . Note that N is the sum of n numbers, and the i th term is the product of all the primes except p i . Show that N cannot be composite, hence N must be prime. Apply the same idea we used in class to deduce a contradiction. 4. [8 points] Let p be a prime number. Show that √ p is irrational. Be sure to explain each step of your argument. 5. [6 points] Find the sum and product of 207 and 114 in Z 23 . 6. [8 points] Use repeated squaring to compute 4 41 (mod 11). 7. [6 points] Compute 231 (mod 37). 8. [6 points] Solve the equation 23 x + 12 = 3 over Z 37 . 9. [6 points] The integer a has the property a 6 ≡ 1 (mod 13). Find ( a 2 )1 (mod 13). Explain!...
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 Fall '08
 Kwong
 Math, Number Theory, Natural number, Prime number, P1 P2 P4, P1 P2 P3

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