Question

For some a and b,HCF of 55 and 210 is 55a + 210b . Find a and b

Answer 1

Let us first find the hcf of 210 and 55
Applying Euclid’s division lemma on 210 and 55 we get
210 = 55 x 3 + 45 . . . . . . . . . . (1)
55 = 45 x 1 + 10 . . . . . . . . . . (2)
45 = 4 x 10 + 5 . . . . . . . . . . (3)
10 = 5 x 2 + 0 Reminder = 0
Hence hcf of (210, 55) = 5
Now
45 = 4 x 10 + 5 [from (3)]
5 = 45 – {10 x 4}
5 = 45 – {[55-45] x4} [from (1)
5 = 45 – 54 x 4 + 45 x 4
5 = 45 x 5 – 55 x 4
5 = {210 – 55 x 3}5 – 55 x 4
5 = 210 x 5 – 55 x 19 [Hcf (210, 55) = 210a + 55y]
Therefore, a = 5 and b = -19