We left off the last installment with the exercise:
To use the Euclidean algorithm on (187,77)
We have (a,b) = (187, 77)
Take:
(187)/ 77 = 2 R33 [e.g. 187 – 154]
= 2*77 + 33
ð (77, 33)
So take:
(77)/ 33 = 2 R11 [e.g. 77 – 66 = 11]
= 2*33 + 11 = (33, 11) = 3*11 + 0
Hence: (187, 77) = (33,11)
Another problem:
Reduce (245, 193) via the Euclidean Algortihm:
(a,b) = (245, 193)
=> (245) / 193 = 1 R52 [e.g. 245 – 193 = 52]
= 1*193 + 52 = (193, 52)
ð (193)/ 52 = 3 R 37 [e.g. 193 – 156 = 33]
= 3*52 + 37 = (52, 37)
But: (52, 37) = (52)/ 37 = 1 R 15 [e.g. 52 – 37 = 15]
= 1*37 + 15 = (37, 15)
But (37, 15) => (37)/ 15 = 2 R 1 = 2*7 + 1
= (7, 1)
7/1 = 7 Hence
(245, 193) = (193, 52) = (52, 37) = (37, 15) = (15, 7) = (7, 1)
Now, let’s look more closely at continued fractions:
We saw in Fig, 1 from the previous blog how 2/5 was converted into a continued fraction.
Now we look at 169/70 and seek the continued fraction development:
Then:
169/70 = 2 + 29/70 = 2 + 1/70/29
70/29 = 2 + 12/29 = 2 + 1/ (29/12)
29/12 = 2 + 5/18 = 2 + 1/ (12/5)
12/5 = 2 + 2/5 = 2 + 1/(5/2)
And: 5/2 = 2 ½ = 2 + ½
Thus 169/70 is developed as shown (Fig. 1) with result = 2.414
Problem: Using the Euclidean Algorithm show the continued fraction for:
(237/ 139)
Show the full continued fraction.
To use the Euclidean algorithm on (187,77)
We have (a,b) = (187, 77)
Take:
(187)/ 77 = 2 R33 [e.g. 187 – 154]
= 2*77 + 33
ð (77, 33)
So take:
(77)/ 33 = 2 R11 [e.g. 77 – 66 = 11]
= 2*33 + 11 = (33, 11) = 3*11 + 0
Hence: (187, 77) = (33,11)
Another problem:
Reduce (245, 193) via the Euclidean Algortihm:
(a,b) = (245, 193)
=> (245) / 193 = 1 R52 [e.g. 245 – 193 = 52]
= 1*193 + 52 = (193, 52)
ð (193)/ 52 = 3 R 37 [e.g. 193 – 156 = 33]
= 3*52 + 37 = (52, 37)
But: (52, 37) = (52)/ 37 = 1 R 15 [e.g. 52 – 37 = 15]
= 1*37 + 15 = (37, 15)
But (37, 15) => (37)/ 15 = 2 R 1 = 2*7 + 1
= (7, 1)
7/1 = 7 Hence
(245, 193) = (193, 52) = (52, 37) = (37, 15) = (15, 7) = (7, 1)
Now, let’s look more closely at continued fractions:
We saw in Fig, 1 from the previous blog how 2/5 was converted into a continued fraction.
Now we look at 169/70 and seek the continued fraction development:
Then:
169/70 = 2 + 29/70 = 2 + 1/70/29
70/29 = 2 + 12/29 = 2 + 1/ (29/12)
29/12 = 2 + 5/18 = 2 + 1/ (12/5)
12/5 = 2 + 2/5 = 2 + 1/(5/2)
And: 5/2 = 2 ½ = 2 + ½
Thus 169/70 is developed as shown (Fig. 1) with result = 2.414
Problem: Using the Euclidean Algorithm show the continued fraction for:
(237/ 139)
Show the full continued fraction.
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