Q. 21

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

Answer :

Given: 468x + 222y


To show: HCF of 468 and 222 as 468x + 222y in two different ways.


Explanation:


HCF of 468 and 222 is found by division method:



Therefore, HCF (468,222) = 6


Now, we need to express the HCF of 468 and 222 as 468x + 222y where x and y are any two integers.


Now, HCF i.e. 6 can be written as,


HCF = 222 - 216 = 222 - (24 × 9)


Writing 468 = 222 × 2 + 24, we get,


HCF = 222 - {(468 222 x 2) × 9}


HCF = 222 - {(468 ×9) (222 × 2 × 9)}


HCF = 222 - (468 × 9) + (222 × 18)


HCF = 222 + (222 × 18) - (468 × 9)


Taking 222 common from the first two terms, we get,


HCF = 222[1 + 18] 468 × 9


HCF = 222 × 19 468 × 9


HCF = 468 × (-9) + 222 × (19)


Let, say, x = -9 and y =19


Then, HCF = 468 ×(x) + 222 ×(y)


Therefore, the HCF of 468 and 222 is written in the form of 468x + 222y where, -9 and 19 are the two integers.


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