Unformatted text preview: x 2 = 11 + y 2 (b) Find all solutions in positive integers to x 2 = 6 + y 2 5. Prove that for any positive integers a and b , ab = gcd( a,b ) Â· lcm( a,b ). Hint: think about prime factorizations. 6. Find all prime numbers p such that p 3 + 3 p is also prime. 7. Prove that a positive integer n is a perfect squre iï¬€ every exponent in its prime factorization is even. 8. (Tricky) Prove that if a â‰¡ b ( mod m ) and a â‰¡ b ( mod n ), then a â‰¡ b ( mod lcm( m,n )). Hint: recall that if m | x and n | x , then lcm( m,n ) | x Diï¬€erent Bases 1. Convert each of the following decimal numbers to the indicated base: (a) 1358 to binary (b) 936 to octal (c) 474 to hexadecimal 2. Convert each of the following to ordinary decimal notation (a) (1101) 2 (b) (347) 8 (c) ( BEEF ) 16...
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- Summer '08
- Math, Prime number, positive integers, Rob Bayer, ordinary decimal notation