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Question
Express the HCF of 468 and 222 as 468x + 222y where x, y are integers in two different ways.
Solution
Given integers are 468 and 222 where 468 > 222.
By applying Euclid’s division lemma, we get 468 = 222 × 2 + 24 …(i)
Since remainder ≠ 0, apply division lemma on division 222 and remainder 24
222 = 24 × 9 + 6 …(ii)
Since remainder ≠ 0, apply division lemma on division 24 and remainder 6
24 = 6 × 4 + 0 …(iii)
We observe that the remainder = 0, so the last divisor 6 is the HCF of the 468 and 222
From (ii) we have
6 = 222 – 24 × 9
⇒ 6 = 222 – [468 – 222 × 2] × 9 [Substituting 24 = 468 – 222 × 2 from (i)]
⇒ 6 = 222 – 468 × 9 – 222 × 18
⇒ 6 = 222 × 19 – 468 × 9
⇒ 6 = 222y + 468x, where x = −9 and y = 19
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