Solution :
Step one – use the Euclidean algorithm
(1) 149 = 1 × 139 + 10
(2) 139 = 13 × 10 + 9
(3) 10 = 1 × 9 + 1
(4) 9 = 9 × 1 + 0
(5)
A remainder of 0 has been obtained, so the gcd is the last non-zero remainder, hence
gcd(149,139) = 1 as required.
Step two – rearrange the earlier expressions to make the remainder the subject.
(3) 1 = 10 – 1 × 9
(2) 9 = 139 – 13 × 10
(1) 10 = 149 – 1 × 139
Step 3 – back substitute
Start with (3) 1 = 10 [× 1] – 1 × 9
Substitute in (2) 1 = 10 × 1 – 1(139 – 13 × 10)
1 = 10 × 1 – 139 × 1 + 13 × 10
1 = 10 × 14 – 139 × 1
Substitute in (1) 1 = (149 × 1 – 1 × 139) × 14 – 139 × 1
1 = 149 × 14 – 14 × 139 – 139 × 1
1 = 149 × 14 – 15 × 139
Thus,149u + 139v = 1 ;where u = 14 and v = –15.
Use Euclidean algorithm to find integers u and v such that 149u + 139V = 1,...
PROBLEM 1 For each of the following pairs of integers, use the Euclidean Algorithm to find ged(a,b), and to write gcd(a,b) as a linear combination of a and b, i.e. find integers m and n such that gcd(a,b) = am + bn. (a) a = 36, b = 60. (b) a = 12628, b = 21361. (c) a = 901, b = -935. (d) a = 72, b = 714. (e) a = -36, b = -60.
To use the Euclidean algorithm to find the greatest counen divisor of each pair of integers' © 2041, 9614 lü) 490256, 674
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