homework7

# homework7 - THEORY OF NUMBERS Math 115 A Homework 7 Due...

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Unformatted text preview: THEORY OF NUMBERS, Math 115 A Homework 7 Due Friday November 30 1. Show that if c 1 , . . . , c φ ( m ) is a reduced system of representatives modulo any m > 2 then c 1 + ··· + c φ ( m ) ≡ (mod m ). 2. Use Euler’s theorem to easily conclude that 4444 4444 ≡ 7 (mod 9). 3. Show that if ( a, 32 760) = 1 then a 12 ≡ 1 (mod 32 760). Hint: decompose 32 , 760 into a product of prime powers and apply the the Chinese Remainder’s Theorem in reverse. 4. Show that if n is not a prime then φ ( n !) n ! = φ (( n- 1)! ( n- 1)! . 5. Characterize all the numbers for which φ ( n ) is a power of 2. 6. Prove that the decimal expression of any rational number is periodic. For this, let a/b be your rational number, with ( a, b ) = 1 and b ∈ N . (a) Show that there are two different non-negative integers k > l ≥ 0 such that (10 k- 10 l ) is a multiple of b . Hint: if ( b, 10) = 1 then apply Euler’s theorem. If not, write b = b 1 b 2 with (10 , b 1 ) = 1 and b 2 = 2 r 1 5 r 2 . Apply Fermat’s little theorem to....
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