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Unformatted text preview: MATH 127: Exam 3 Review Monday, April 18, 2011 1. Show that the square of every odd integer is of the form 8 k + 1 2. Show that if a,b Z , not both 0, and c Z with c 6 = 0 then ( ca,cb ) = | c | ( a,b ) 3. Show that if k N then (3 k + 2 , 5 k + 3) = 1 4. Use the Euclidean Algorithm to find the gcd of 1776 and 1492 and then express this gcd as a linear combination of 1776 and 1492 5. Show that if a,b N and a 3 | b 2 then a | b 6. For p prime, we say p a strongly divides n , denoted p a || n iff p a | n but p a +1 6 | n (a) Show that if p a || m and p b || n then p a + b || n (b) Show that if p a || m and p b || n with a 6 = b then p min ( a,b ) || ( m + n ) 7. Show that if a,b,c Z and c | ab then c | ( a,c )( b,c ) 8. Show that if a,b,c N with ( a,b ) = 1 and ab = c n then there exists d,e N with a = d n , b = e n 9. For the following diophantine equations, either find all solutions or show that there are none (a) 30 x + 47 y =- 11 (b) 25 x + 95 y = 970 10. A vegetarian banquet is offering a seitan dish and a tofu dish. The seitan costs $11 and the10....
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This note was uploaded on 09/08/2011 for the course MATH 21-127 taught by Professor Gheorghiciuc during the Spring '07 term at Carnegie Mellon.
- Spring '07