MAT445 notes

MAT445 notes - G1NTH: Introduction to Number Theory Problem...

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G1NTH: Introduction to Number Theory Problem Class 5 (Solutions) 1. Find the last digit of the decimal expansion of 3 1000 . Solution. The last digit of the decimal expansion of 3 1000 is the remainder of the division of 3 1000 by 10 . Since φ (10) = 4 , Euler’s theorem implies that 3 1000 = (3 4 ) 250 1 mod 10 . Therefore, the last digit is 1 . 2. Show that σ ( n ) is an odd integer if and only if n is a perfect square or twice a perfect square. Solution. If n = p n 1 1 . . . p n s s then σ ( n ) = (1 + p 1 + ··· + p n 1 1 ) . . . (1 + p s + ··· + p n s s ) . This is odd if and only if each factor is odd. Now, if p 1 = 2 , then 1 + p 1 + ··· + p n 1 1 is always odd. If p i is an odd prime, 1 + p i + ··· + p n i i is odd if and only if n i is even. Therefore, σ ( n ) is odd if and only if either n is a perfect square (in the case 2 appears raised in an even power in the prime decomposition of n ) or twice a perfect square (otherwise). 3. If the ciphertext message produced by RSA encryption with key (
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This note was uploaded on 11/29/2010 for the course MAT 23358 taught by Professor Donjones during the Spring '10 term at ASU.

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MAT445 notes - G1NTH: Introduction to Number Theory Problem...

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