Express the HCF of 468 and 222 as 468x + 222y where x, y are integers in two different ways.
Answers
Answer:
The HCF of 468 and 222 as 468x + 222y is 6 and in 1st way x = - 9 & y = 19 and in 2nd way x = 213 and y= - 449
Step-by-step explanation:
SOLUTION :
By Euclid’s division algorithm,a = bq + r
Here, a > b , a = 468 & b = 222
HCF of 468 and 222
468 = (222 x 2) + 24 ……………….(1)
222 = (24 x 9) + 6 …………………..(2)
24 = (6 x 4) + 0
Here Remainder is 0 , so
HCF of 468 and 222 is 6
Hence, the HCF of 468 and 222 is 6.
Ist way :
From eq 2 , we get
6 = 222 - (24 x 9)
6 = 222 - (468 – 222 x 2) x 9
[24 as (468 – 222 x 2) from eq 1 ]
6 = 222 - (468 x 9 – 222 x 2 x 9)
6 = 222 - (468 x 9) + (222 x 18)
6 = 222 + (222 x 18) - (468 x 9)
6 = 222 (1 + 18) – 468 x 9
[By taking 222 as common]
6 = 222 x 19 – 468 x 9
6 = 468 x - 9 + 222 x 19
Therefore, HCF of 468 and 222 is (468 x -9 + 222 x 19) in the form 468x + 222y, where x = - 9 & y = 19
Hence, x = - 9 & y = 19
2nd way :
HCF (d) of two positive integers a and b can be expressed as a linear combination of a and b ,i.e., d = xa + yb for some integer x and y.
d = xa + yb
d = xa + yb + ab - ab
d = xa + ab + yb -ab
d = a(x+ b)+ b(y- a)
From 1st way we have :
HCF(d) = 6 , a = 468 ,b = 222, x = - 9 y = 19
6 = 468 (- 9 + 222) + 222(19 - 468)
6 = 468(213) + 222(- 449)
Therefore, HCF of 468 and 222 is (468 x 213 + 222 x - 449) in the form 468x + 222y, where x = 213 & y = - 449
Hence, in 1st way x = - 9 & y = 19 and in 2nd way x = 213 and y= - 449
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Step-by-step explanation:
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