Description
see the file
Unformatted Attachment Preview
Purchase answer to see full attachment
Explanation & Answer
Attached.
Surname 1
Name
Supervisor
Course
Date
1. a) let 𝑎 = 513 , 𝑏 = 187 then the (513, 187) is calculated as follows:
513 = 2.187 + 139
187 = 1.139 + 48
139 = 2.48 + 43
48 = 1.43 + 5
43 = 8.5 + 3
5 = 1.3 + 2
3 = 1.2 + 1
2 = 1.1 + 1 Thus ( 513,187) = 1
For us to find the linear combination we proceed as follows:
𝑎 = 2𝑏 + 139 ⟹ 𝑎 − 2𝑏 = 139
𝑏 = 𝑎 − 2𝑏 + 48 ⟹ 3𝑏 − 𝑎 = 48
𝑎 − 2𝑏 = 2(3𝑏 − 𝑎) + 43 ⟹ 3𝑎 − 8𝑏 = 43
3𝑏 − 𝑎 = 3𝑎 − 8𝑏 + 5 ⟹ 11𝑏 − 4𝑎 = 5
3𝑎 − 8𝑏 = 8(11𝑏 − 4𝑎) + 3 ⟹ 35𝑎 − 96𝑏 = 3
11𝑏 − 4𝑎 = 35𝑎 − 96𝑏 + 2 ⟹ 107𝑏 − 39𝑎 = 2
35𝑎 − 96𝑏 = 107𝑏 − 39𝑎 + 1 ⟹ 74𝑎 − 203𝑏 = 1
Thus the linear combination is:74.513 − 203.187 = 1
Surname 2
1.b) let 𝑎 = 84 , 𝑏 = 54 then gcd(84,54) is
84 = 1.54 + 30
54 = 1.30 + 24
30 = 1. 24 + 6
24 = 5.6 + 0 ; Thus the gcd(84,54) = 6
We can write as a linear combination as follows:
𝑎 = 𝑏 + 30 ⟹ 𝑎 − 𝑏 = 30
𝑏 = 𝑎 − 𝑏 + 24 ⟹ 2𝑏 − 𝑎 = 24
𝑎 − 𝑏 = 2𝑏 − 𝑎 + 6 ⟹ 2𝑎 − 3𝑏 = 6
Hence the linear combination is: 2.84 − 3.54 = 6
1.c) let 𝑎 = 6540, 𝑏 = 1206 then (6540,1206) is:
6540 = 5. 1206 + 510
1206 = 2.510 + 186
510 = 2.186 + 138
186 = 1. 138 + 48
138 = 2.48 + 42
48 = 1.42 + 6
42 = 7.6 + 0 ℎ𝑒𝑛𝑐𝑒 (6540,1206) = 6
Thus the linear combination of (6540,1206) is written as
𝑎 = 5𝑏 + 510 ⟹ 𝑎 − 5𝑏 = 510
𝑏 = 2(𝑎 − 5𝑏) + 186 ⟹ 11𝑏 − 2𝑎 = 186
𝑎 − 5𝑏 = 2(11𝑏 − 2𝑎) + 138 ⟹ 5𝑎 − 27𝑏 = 138
Surname 3
11𝑏 − 2𝑎 = 5𝑎 − 27𝑏 + 48 ⟹ 38𝑏 − 7𝑎 = 48
5𝑎 − 27𝑏 = 2(38𝑏 − 7𝑎) + 42 ⟹ 19𝑎 − 103𝑏 = 42
19𝑎 − 103𝑏 = 7.6 𝑡ℎ𝑢𝑠 6 =
Hence 6 =
1
7
1
(19𝑎 − 103𝑏)
7
(19.513 − 103.187)
2.) We know that if 𝑎│𝑏 then 𝑎𝑘 = 𝑏 for some integer 𝑘 and If 𝑎 │𝑐 then 𝑎𝑚 = 𝑐 for some
integer 𝑚
Thus 𝑏𝑐 = (𝑎𝑘)(𝑎𝑚) = 𝑘𝑚𝑎2 = 𝑛 𝑎2 for some 𝑘𝑚 = 𝑛 ∈ 𝑧. Hence 𝑎2 │𝑏𝑐
3.) We find the gcd(21,15) and then find the numbers between 0 and 10 with the found 𝑔𝑐𝑑
Thus (21,15) = 3 and thus 0, 1, 3, 6 and ...