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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online