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Unformatted text preview: , 59 or 118 . 4. For m ≥ 2, show that [ m1] is always a unit in the ring Z /m Z . What is its order? Solution: [ m1] m = [1] m so [ m1]1 m = [ m1] m . If m=2, its order is 1; otherwise it is 2. 5. Show that for any integer n , 21 divides n 25n . Solution: Show both 3 and 7 divide n 25n = n ( n 241) . By Fermat, either 3  n or n 2 ≡ 1(mod 3) , in which case n 24 = ( n 2 ) 12 ≡ 1(mod 3) . Either way, 3  n ( n 241) . Similarly, either 7  n or n 6 ≡ 1(mod 7) , in which case n 24 = ( n 6 ) 4 ≡ 1(mod 7) . Again, we conclude 7  n ( n 241) . 1...
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This note was uploaded on 02/24/2008 for the course MATH 3360 taught by Professor Billera during the Spring '08 term at Cornell University (Engineering School).
 Spring '08
 BILLERA
 Math, Algebra

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