The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be expressed as a unique product of prime numbers. To find the highest common factor (HCF) of the numbers 26, 51, and 91, we’ll first break each number down into its prime factorization.
1. **Finding Prime Factorizations:**
- 26: The prime factorization of 26 is 2 × 13.
- 51: The prime factorization of 51 is 3 × 17.
- 91: The prime factorization of 91 is 7 × 13.
2. **Identifying Common Factors:**
Next, we need to find the common factors from these prime factorizations:
- 26: 2, 13
- 51: 3, 17
- 91: 7, 13
From the list above, we can see that the only prime factor that appears in multiple factorizations is 13.
3. **Calculating the HCF:**
Thus, the highest common factor (HCF) of 26, 51, and 91 is: 13.
In conclusion, by applying the Fundamental Theorem of Arithmetic, we found that the HCF of the numbers 26, 51, and 91 is 13.