Question

Express the HCF of number 72 and 124 as a linear combination of 72 and 124

Answer 1

Solution:-
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)

Answer 2

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)