Explanation:

To find the Highest Common Factor (HCF) of three numbers, we can use various methods such as prime factorization, division method, and Euclid's algorithm. Here, we will use the prime factorization method to find the HCF of 45, 64, and 27.

Prime Factorization:

Prime factorization involves breaking down each number into its prime factors and then finding the common factors among them.

Step 1: Prime factorization of 45:
- 45 can be divided by 3, so we have the factors 3 and 15.
- 15 can be divided by 3, so we have the factors 3 and 5.
- 5 is a prime number, so we stop here.
- The prime factors of 45 are 3 * 3 * 5 = 3^2 * 5.

Step 2: Prime factorization of 64:
- 64 can be divided by 2, so we have the factors 2 and 32.
- 32 can be divided by 2, so we have the factors 2 and 16.
- 16 can be divided by 2, so we have the factors 2 and 8.
- 8 can be divided by 2, so we have the factors 2 and 4.
- 4 can be divided by 2, so we have the factors 2 and 2.
- 2 is a prime number, so we stop here.
- The prime factors of 64 are 2 * 2 * 2 * 2 * 2 = 2^6.

Step 3: Prime factorization of 27:
- 27 can be divided by 3, so we have the factors 3 and 9.
- 9 can be divided by 3, so we have the factors 3 and 3.
- 3 is a prime number, so we stop here.
- The prime factors of 27 are 3 * 3 * 3 = 3^3.

Common Factors:

Now that we have the prime factorization of each number, we can find the common factors by selecting the minimum power of each common prime factor.

The common prime factors among 45, 64, and 27 are 3 and 3. However, the power of 3 in 45 is 2, in 64 is 0, and in 27 is 3. We select the minimum power of 3, which is 0.

HCF:

To find the HCF, we multiply the common prime factors with the minimum powers:
HCF = 3^0 = 1

Therefore, the HCF of 45, 64, and 27 is 1.

45=3×3×5×1
64=2×2×2×2×2×2×1
27=3×7×1. so,HCF=1