Question

In: Math

A. 271 and 516 1. Find the greatest common divisor, d, of the two numbers from part...

A. 271 and 516

1. Find the greatest common divisor, d, of the two numbers from part A using the Euclidean algorithm. Show and explain all work.

2. Find all solutions for the congruence ax ? d (mod b) where a and b are the integers from part A and d is the greatest common divisor from part A1. Show and explain all work.

Solutions

Expert Solution

271 and 516

Step 1

A=271, B=516
A ?0
B ?0
Use long division to find that 516/271 = 1 with a remainder of 245. write this as: 516 = 271 * 1 + 245
Find GCD(271,245), since GCD(516,271)=GCD(271,245)

A=271, B=245

Step 2

A ?0
B ?0
Use long division to find that 271/245 = 1 with a remainder of 26. Write this as: 271 = 245 * 1 + 26
Find GCD(245,26), since GCD(271,245)=GCD(245,26)

A=245, B=26

Step 3

A ?0
B ?0
Use long division to find that 245/26 = 9 with a remainder of 11. Write this as: 245 = 26 * 9 + 11
Find GCD(26,11), since GCD(245,26)=GCD(26,11)

A=26, B=11

Step 4

A ?0
B ?0
Use long division to find that 26/11 = 2 with a remainder of 4. Write this as: 26 = 11 * 2 + 4
Find GCD(11,4), since GCD(26,11)=GCD(11,4)

A=11, B=4

Step 5

A ?0
B ?0
Use long division to find that 11/4 = 2 with a remainder of 3. Write this as: 11 = 4 * 2 + 3
Find GCD(4,3), since GCD(11,4)=GCD(4,3)

A=4, B=3

Step 6

A ?0
B ?0
Use long division to find that 4/3 = 1 with a remainder of 1. Write this as: 4 = 3 * 1 + 1
Find GCD(3,1), since GCD(4,3)=GCD(3,1)

A=3, B=1

Step 7

A ?0
B ?0
Use long division to find that 3/1 = 3 with a remainder of 0. Write this as: 3 = 3 * 1 + 0
Find GCD(1,0), since GCD(3,1)=GCD(1,0)

A=1, B=0

Step 8

A = 0
B ?0

so GCD (516,271) = 1

milcah answered 1 year ago

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