Question

For each of the following pairs of numbers, find the gcd of the two numbers, and express the gcd as a linear combination of the two numbers. (a) 56 and 42 (b) 81 and 60

Answer 1

Answer:

a. 14 and 14= (-1)56+1(44)

b. 3 and 3= (-4)60+3(81)

Step-by-step explanation:

Case 1: GCD is the highest common divisor for both numbers without reminder

a. GCD of 56 and 42 = 14

b GCD of 81 and 60 = 3

Case 2: Express GCD as a Linear combination.

a. GCD = 14 which is also the difference between 56 and 42

Therefore

14= (-1)56+1(44) expresses 14 as a linear combination of 56 and 44.

b. GCD=3, we need to look for a multiple of 60 and 81 that their differences would be 3

Therefore the multiples are:

For 60 = 240 (60×4)

For 81 = 243 (81×3)

3= (-4)60+3(81) expresses 3 as a linear combination of 81 and 60