Express the H.C.F of number 72 and 124 as a linear combination of 72 and 124.

First we will find the find the H.C.F of numbers 72 and 124.

72 = 2 × 2 × 2 × 3 × 3

⇒ 2³ × 3²

124 = 2 × 2 × 31

⇒ 2² × 31

The H.C.F of these numbers product of the least power of each common factor

Here 2 is the only common factor and least power of which is 2.

So, we have

H.C.F (72 and 124) = 2 × 2

= 4

Now, to express H.C.F = 4 as a linear combination of 72 and 124.

⇒ 4 = 72a + 124b, where a and b are integers. (By Euclid division Lemma)

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

H.C.F (72 and 124) = 4

= 72(–12) + 124(7)

So 4 = 72a + 124b, where a = – 12 and b = 7