Question

HCF of 130 and 42 is 2. So how can we prove the reverse order ie,2 is the HCF of 130 and 42

Answer 1

130=42*3 + 4, 4 is the remainder.

Apply Euclid's division lemma to 42 and 4

42= 4* 10 +2, 2 is the remainder

Apply Euclid's division lemma to 4 and 2

4 = 2*2 +0, where 0 is the remainder

So, 2 is the HCF.

Proving 2 as HCF:

2=42-4*10

=42-[130-3*42]10

=42-130*10 + 3*42*10

=42-130*10 + 30*42

= 42(1+30)+130(-10)

=42(31)+130(-10)

= 42 x+130y where x=31 and y=-10

[If d= ax+by for some integers x and y then d = HCF(a,b)]

So, 2 = HCF(130,42)