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hw_02

# hw_02 - k there exist k consecutive integers each of which...

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Number Theory, Homework 2nd batch due: Wednesday, February 4 All exercises are worth 5 points. Please select five problems since there is a cap of 25 points. If you decide to do more than five problems, please indicate which ones you want graded since otherwise the five lowest scores will be selected. Sometimes, only five problems will be assigned, in which case the selection issue is moot. Exercise 1. 2.3.3: Solve: x 1(4), X 0(3), and x 5(7). Exercise 2. 2.3.9: Determine (with proof) the values of n for which the Euler func- tion φ ( n ) is odd. Exercise 3. 2.3.17: Solve the congruence x 3 - 9 x 2 + 23 x - 15 0(143) . Exercise 4. 2.3.18: Show that for any positive integer
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Unformatted text preview: k there exist k consecutive integers each of which is divisible by a square. Exercise 5. 2.3.36: Find the last two decimal digits of 2 1000 and 3 1000 . Exercise 6. 2.6.2: Solve the congruence x 5 + x 4 + 1 ≡ 0(3 4 ) . Exercise 7. 2.6.11: Let p be a prime and f ( x 1 , x 2 , . . . , x n ) be an integer polynomial in n variables, and suppose that f ( a 1 , . . . , a n ) ≡ 0( p ) and that the partial derivative ∂f ∂x i ( a 1 , . . . , a n ) ±≡ 0( p ) . Show that the congruence f ( x 1 , . . . , x n ) ≡ 0( p j ) has a solution for each j ....
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