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Unformatted text preview: (unless noted otherwise). (7) Be careful to avoid making grievous errors that are subject to heavy penalties. (8) If you need more blank paper, ask a proctor. Out of fairness to others, please stop working and close the exam as soon as the time is called. A signiﬁcant number of points will be taken oﬀ your exam score if you continue working after the time is called. You will be given a twominute warning before the end. 1 2 3 4a 4b 4c 5 Total 1 1. (15 points) Find the prime decomposition of the number 1! · 2! · 3! · 4! · 5! · 6! 2. (20 points) Compute 19 37 mod 12. 2 3. (10 points) Find the maximal power of 5 that divides 100! 4. a) (15 points) Solve the congruence 28 x = 7 mod 63. 3 b) (20 points) Solve the system of congruences x = 7 mod 10 , x = 13 mod 19 . c) (10 points) Find the greatest common divisor of 313 and 515. Justify your answer. 4 5. (10 points) Prove the identity 1 · 2 · 3 + 2 · 3 · 4 + ... + n · ( n + 1) · ( n + 2) = n ( n + 1)( n + 2)( n + 3) 4 . 5...
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 Summer '08
 BESCHER
 Algebra, Prime number, Greatest common divisor, Divisor, congruences

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