Question

Find the HCF of the following pairs of integers and express it as a linear combination of 963 and 657.

Answer 1

By applying Euclid’s division lemma 963 = 657 × 1 + 306 …(i)

Since remainder ≠ 0, apply division lemma on divisor 657 and remainder 306

657 = 306 × 2 + 45 ….. (ii)

Since remainder ≠ 0, apply division lemma on divisor 306 and remainder 4

306 = 45 × 6 + 36 …..(iii)

Since remainder ≠ 0, apply division lemma on divisor 45 and remainder 36

45 = 36 × 1 + 9 …… (iv)

Since remainder ≠ 0, apply division lemma on divisor 36 and remainder 9

36 = 9 × 4 + 0

∴ HCF = 9

Now 9 = 45 – 36 × 1 [from (iv)]

= 45 – [306 – 45 × 6] × 1 [from (iii)]

= 45 – 306 × 1 + 45 × 6

= 45 × 7 – 306 × 1

= 657 × 7 – 306 × 14 – 306 × 1 [from (ii)]

= 657 × 7 – 306 × 15

= 657 × 7 – [963 – 657 × 1] × 15 [from (i)]

= 657 × 22 – 963 × 15