hw_01

# hw_01 - 1.2.31 Let n ≥ 2 and k be any positive integers...

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Number Theory, Homework 1st batch due: Wednesday, January 28 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. 1.2.2: Find the greatest common divisor g of the numbers 1819 and 3587 as well as integers x and y satisfying 1819 x + 3587 y = g. Exercise 2. 1.2.9: Show that if ac | bc then a | b . Exercise 3.
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Unformatted text preview: 1.2.31: Let n ≥ 2 and k be any positive integers, Prove that ( n-1) | ( n k-1) . Exercise 4. 1.2.32: Let n ≥ 2 and k be any positive integers, Prove that ( n-1) 2 | ( n k-1) if and only if ( n-1) | k . Exercise 5. 1.3.24: Show that n ≥ 2 has a prime factor p ≤ √ n if n is composite. Exercise 6. 1.3.36: Let S be the set of integers 1 , 2 , 3 , . . . , n and let 2 k be the highest power of 2 within S . Prove that 2 k does not divide any other element of S . Deduce that ∑ n i =1 1 i is not an integer for n ≥ 2....
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