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Express the numbers 72 and 124 as linear combination of 72 and 124. Please give me the answer by method and not by hit and trial method. Explain the steps which follow.

Express the numbers 72 and 124 as linear combination of 72 and 124. Please give me the answer by method and not by hit and trial method. Explain the steps which follow.

Grade:10

1 Answers

Arun
25758 Points
5 years ago
Consider the following numbers
72 and 124
First find the HCF of above numbers.
72 = 2 × 2 × 2 × 3 × 3
 ⇒  2³ × 3²
124 = 2 × 2 × 31
⇒  2² × 31
To find the HCF of these numbers, take the least power of each common factor and find the product.
Here 2 is the only common factor and least power of which is 2.
So, we have
HCF(72 and 124) = 4
Now we need to express HCF = 4 as a linear combination of 72 and 124.
That is,
4 = 72a + 124b, where a and b are integers.
Use hit and trial method.
Take a = -10 and b = 6
72(-10) + 124(6) 
⇒ -720 + 744 = 24, which is not equal to 4.
So, take a = -12 and b =7
72(-12) + 124(7)
⇒  - 864 + 868 = 4
So, the required linear combination is
HCF(72 and 124) = 4 = 72(-12) + 124(7)

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