Express the HCF of 468 and 222 as 468x + 222y in Two Different Ways

Video Explanation

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Solution

Question: Express the HCF of 468 and 222 as 468x + 222y, where x and y are integers, in two different ways.

Step 1: Find the HCF of 468 and 222

Using Euclid’s division algorithm:

468 = 222 × 2 + 24

222 = 24 × 9 + 6

24 = 6 × 4 + 0

Since the remainder is zero,

∴ HCF (468, 222) = 6

Step 2: Express HCF as a Linear Combination (First Way)

From the above steps:

6 = 222 − 24 × 9

6 = 222 − (468 − 222 × 2) × 9

6 = 222 − 9 × 468 + 18 × 222

6 = 19 × 222 − 9 × 468

∴ 6 = −9 × 468 + 19 × 222

Step 3: Express HCF as a Linear Combination (Second Way)

Another solution can be obtained by adding suitable multiples:

∴ 6 = 28 × 468 − 59 × 222

Final Answer

The HCF of 468 and 222 is 6 and it can be expressed as:

6 = −9 × 468 + 19 × 222

6 = 28 × 468 − 59 × 222

Conclusion

Thus, the HCF of 468 and 222 can be expressed as a linear combination of the given numbers in two different ways using Euclid’s division algorithm.