Hmk3sol - COT 3100 Homework # 3 Solutions 1) Use Euclids...

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1) Use Euclid’s Algorithm to find the greatest common divisor(GCD) of 87 and 57. 87 = 1*57 + 30 57 = 1*30 + 27 30 = 1*27 + 3 27 = 9*3, so gcd(87,57) = 3. 2) Show that if 7 | 3x + 4y, then there are no integer solutions to the equation 23x + 26y = 20000 23x + 26y = 14x + 9x + 14y + 12y = 14x + 14y + 9x + 12y = 14(x + y) + 3(3x + 4y) But we know that 7 | 3x + 4y, so we can let 7d = 3x + 4y, where d is an integer. = 14(x+y) + 3(7d) = 7(2(x+y) + 3d) Thus, we find that 7 | (23x + 26y). But, on the other hand, 7 is NOT a factor of 20000. So, it is impossible for the LHS and RHS of the given equation to be equal. 3) Prove that if 3 | x + 4y, then 12 | 20x + 44y. 20x + 44y = 12x + 8x + 12y + 32y = 12x + 12y + 8x + 32y = 12(x + y) + 8(x + 4y) But, we know that 3 | x + 4y, so we can let 3d = x + 4y, where d is an integer. = 12(x+y) + 8(3d) = 12(x+y) + 12(2d) = 12(x+y+2d) Thus, we have that 12 | (20x + 44y). 4) Using mod rules, find the remainder when you divide 7 200 by 9. 7
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Hmk3sol - COT 3100 Homework # 3 Solutions 1) Use Euclids...

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