Obtain the H.C.F of 420 and 272 by using Euclid 's division algorithm and verify the same by using the fundamental theorem of arithmetic.

To find the largest number which divides 650 and 1170, we need to find the Highest Common Factor. According to Euclid’s Division Lemma, if a and b are any two positive integers then there exist two unique whole numbers q and r such that

a = b q + r,

where 0 ≤ r < b

Here, a is called the dividend,

b is called the divisor,

q is called the quotient and

r is called the remainder.

So, apply the lemma on 420 and 272.

We get,

420 = (272 × 1) + 148

Since the remainder is not zero.

Apply the lemma again on 272 and 148.

We get,

270 = (148 × 1) + 122

Since the remainder is not zero.

Apply the lemma again on 148 and 122.

We get,

148 = (122 × 1) + 26

Since the remainder is not zero.

Apply the lemma again on 122 and 26.

We get,

122 = (26 × 4) + 18

Since the remainder is not zero.

Apply the lemma again on 26 and 18.

We get,

26 = (18 × 1) + 8

Since the remainder is not zero.

Apply the lemma again on 18 and 8.

We get,

18 = (8 × 2) + 2

Since the remainder is not zero.

Apply the lemma again on 8 and 2.

We get,

8 = (2 × 4) + 0

We have finally got remainder as 0.

⇒ HCF (420, 272) = 2

By Fundamental Theorem of Arithmetic, we get

The prime factorization of 420 and 272 are:
420 = 2 × 2 × 3 × 5 × 7
272 = 2 × 2 × 2 × 2 × 17

∴ HCF (470, 272) = 2 × 2 = 4