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Q.

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


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a

6=(468)×(−9)+222×(18);6=468×(213)+222×(−449)  

b

6=(468)×(−9)+222×(19);6=468×(213)+222×(−448)  

c

6=(468)×(−9)+222×(19);6=468×(213)+222×(−449)  

d

6=(468)×(9)+222×(19);6=468×(213)+222×(−449)   

answer is C.

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Detailed Solution

Given to find the HCF of two integers 468 and 222 and to express it as a linear combination as 468 x+222 y where x, y are integers.
Euclid's Division Lemma states that if two positive integers a and b exist, then there must be unique values of q and r that satisfy the formula a= bq + r, where  0 ≤ r < b.
Apply Euclid’s division lemma on 468,222
468=222×2+24   ………….(1)
∵  remainder ≠ 0,  apply Euclid’s division lemma on divisor 222 and remainder 24
222=24×9+6   …………….. (2)
∵  remainder ≠ 0,  apply Euclid’s division lemma on divisor 24 and remainder 6
  24=6×4+0  …………..(3)
∴  remainder = 0  
∴  HCF of 468, 222 is 6.
Now, equation (2) can be expressed as,
6=222−(24×9)  …………..(4)
Substitute (1) in (4),
6 =222−[(468−222×2)×9] =222−[(468×9)−(222×2×9)] =222−(468×9)+(222×18) =222+(222×18)−(468×9).................(5)  
Take 222 as common from (5)
6 =222(1+18)−(468×9) =222×19−(468×9) =(468)×(−9)+222×(19).................(6)  
Above equation is in the form, 6=468×(x)+222×(y)  with x = -9 and y = 19.
Therefore this is the one way to express HCF of 468, 222.
Now, (6) can also rewrite as,
6 =468×(−9)+222×(19)+468(222)−222(468) =468×(213)+222×(−449)  
Above equation is in the form, 6=468 x +222 y  with x = 213 and y = - 449.
∴  HCF of 468 and 222 is 6 and can be expressed as a linear combination of 468 and 222 in two different ways as,
6=(468)×(−9)+222×(19)  
6=468×(213)+222×(−449)  
Hence the correct option is 3.     
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