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Q.

Find the HCF of 963 and 657 and express it as a linear combination of them.


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a

9;9 = 657 × 22 – 963 × 15

b

8;8 = 657 × 22 – 963 × 15

c

7;7 = 657 × 22 – 963 × 15

d

6;6 = 657 × 22 – 963 × 15 

answer is A.

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Detailed Solution

Given to find the HCF of two integers 963 and 657 and to express it as linear combination of 963 and 657.
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 963, 657
963=657×1+306   . . .  . . .(1)
∵  remainder ≠ 0,  apply Euclid’s division lemma on divisor 657 and remainder 306
657=306×2+45   . . . . . . (2)
∵  remainder ≠ 0,  apply Euclid’s division lemma on divisor 306 and remainder 45
306=45×6+36   . . . . . . (3)
∵  remainder ≠ 0,  apply Euclid’s division lemma on divisor 45 and remainder 36
45=36×1+9   . . . . . . .(4)
∵  remainder ≠ 0,  apply Euclid’s division lemma on divisor 36 and remainder 9
36=9×4+0   . . . . . .(5)
∴  remainder = 0  
∴  HCF of 963, 657 is 9
Now, equation (4) can be expressed as,
9=45−36×1   . . . . . .(6)
Substitute (3) in (6)
9=45−(306−45×6)×1     ⇒45−306×1+45×6   . . . . . . (7)
 Substitute (2) in (7)
9=(657−306×2)×7−306×1  
⇒657×7−306×14−306×1   ⇒657×7−306×15……(8)  
 Substitute (1) in (8)
  9=657×7−(963−657×1)×15 ⇒657×7−963×15+657×15 ⇒657×22−963×15  
∴  HCF of 963 and 657 is 9 and can be expressed as a linear combination of 963 and 657 by,
9 = 657 × 22 – 963 × 15.
Hence the correct option is 1.
 
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