The HCF of 420 and 272 is 4.
To find the HCF of 420 and 272 using Euclid's division algorithm, we can proceed as follows:
Step 1: Divide 420 by 272
420 = 272 x 1 + 148
Step 2: Divide 272 by 148
272 = 148 x 1 + 124
Step 3: Divide 148 by 124
148 = 124 x 1 + 24
Step 4: Divide 124 by 24
124 = 24 x 5 + 4
Step 5: Divide 24 by 4
24 = 4 x 6 + 0
So, we can see that the last non-zero remainder obtained by Euclid's division algorithm is 4. Therefore, the HCF of 420 and 272 is 4.
Now, let's verify this using the Fundamental Theorem of Arithmetic.
The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 can be expressed as a product of prime numbers in a unique way, up to the order of the factors.
The prime factorization of 420 is:
420 = 2^2 x 3 x 5 x 7
The prime factorization of 272 is:
272 = 2^4 x 17
To find the HCF using the prime factorizations, we need to take the product of the common prime factors with the smallest exponent. In this case, the only common prime factor is 2, and it appears with an exponent of 2 in 420 and an exponent of 4 in 272. So, the HCF is:
HCF(420, 272) = 2^2 = 4
This matches the result obtained using Euclid's division algorithm. Therefore, we have verified that the HCF of 420 and 272 is indeed 4.