Solve for a.

ab+c=d

1. a=b/(c-d)
2. a=(d-c)/b
3. a=b+c/d

Let d = (110257, 110385). We know that d has to divide the difference of the

numbers: d|(110385 − 110257) = 128. We note that 128 = 27

, which means that d

is either 1 or is a power of 2. However, both 110257 and 110385 are odd, hence d

cannot be even. It follows that d = 1 and that 110257

110385 is reduced.

(2) Suppose b = (a, d). Note that

a = qd + r ⇒ r = a − qd.

Since b|d and b|a, we get b|r. Then b is a common divisor of both d and r, implying

that b|(r, d). Since (r, d) is the greatest common divisor, we get

b = (a, d) 6 (r, d).

Similarly, let c = (r, d). Then c|a as a = qd + r. We get that c is a common

divisor of both a and d, implying that c|(a, d). Then we get

c = (r, d) 6 (a, d).

As a result, we get

(a, d) = (r, d).

(3) (i) a = 37, b = 50. By Euclidean Algorithm

50 = 37 + 13

37 = 2(13) + 11

13 = 11 + 2

11 = 5(2) + 1

2 = 2 + 0.

Therefore, (37, 50) = 1.

(ii) a = 2709, b = 5518. By Euclidean Algorithm

5518 = 2(2709) + 100

2709 = 27(100) + 9

100 = 11(9) + 1

9 = 9(1) + 0.

Therefore, (5518, 2709) = 1.

(4) By using 1.3 (i)

1 = 11 − 5(2) = 11 − 5(13 − 11) = 6(11) − 5(13) = 6(37 − 2(13)) − 5(13)

= 6(37) − 17(13) = 6(37) − 17(50 − 37) = 23(37) − 17(50).

Hence x = 23 and y = −17.

Step-by-step explanation:

a

step-by-step explanation:

step-by-step explanation:

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