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Q.

Find the HCF of 1288 and 575 and express it as a linear combination of them.


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a

24;24 =  575 × 9 - 1288 × 4

b

23;23 =  575 × 9 - 1288 × 4

c

22;22 =  575 × 9 - 1288 × 4

d

25;25 =  575 × 9 - 1288 × 4 

answer is B.

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Detailed Solution

Given to find the HCF of two integers 1288 and 575 and to express it as a linear combination of 1288 and 575.
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 1288, 575    . . .  . . .(1)
1288=575×2+138   . . .  . . .(1)
∵  remainder ≠ 0,  apply Euclid’s division lemma on divisor 575 and remainder 138
575=138×4+23   . . . . . . (2)
∵  remainder ≠ 0,  apply Euclid’s division lemma on divisor 138 and remainder 23
138=23×6+0   . . . . . . (3)
∴  remainder = 0  
Hence, HCF of 1288, 575 is 23
Now, from (3)
23=575−138×4   . . . . . .(4)
Substitute (1) in (4),
23=575−(1288−575×2)×4       =575−1288×4+575×8       =575×9−1288×4  
∴  HCF of 1288 and 575 is 23 and can be expressed as a linear combination of 1288 and 575 by,
23 =  575 × 9 - 1288 × 4.
Hence the correct option is 2.
 
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