What are common prime factor of 77 and 242?
The Correct Answer and Explanation is:
Common prime factors of 77 and 242:
First, let’s find the prime factorization of both numbers:
- 77:
77 can be divided by 7 (which is a prime number).
77÷7=1177 \div 7 = 11 (11 is also a prime number).
So, prime factors of 77 are 7 and 11. - 242:
242 is even, so start dividing by 2:
242÷2=121242 \div 2 = 121
121 is 11×1111 \times 11, so its prime factors are 11 and 11 (or 11211^2).
So, prime factors of 242 are 2 and 11 (since 11 repeats twice).
Common prime factors:
Looking at the prime factors:
- 77 = 7×117 \times 11
- 242 = 2×11×112 \times 11 \times 11
The common prime factor is 11.
Explanation
Prime factors are the basic building blocks of any integer. They are prime numbers that multiply together to give the original number. Every integer greater than 1 can be uniquely expressed as a product of prime factors. This is known as the Fundamental Theorem of Arithmetic.
When we want to find the common prime factors of two numbers, we are essentially looking for prime numbers that appear in both factorizations. These common prime factors reveal the shared “building blocks” between the two numbers and are useful in various applications such as simplifying fractions, finding greatest common divisors (GCD), or solving problems related to divisibility.
In this case, we analyzed two numbers: 77 and 242.
- The number 77 factors into 7 and 11, both prime numbers.
- The number 242 factors into 2 and 11 twice (since 121 = 11 × 11).
The only prime number appearing in both lists is 11. Hence, 11 is the common prime factor.
Why is this important? Knowing the common prime factors allows us to calculate the greatest common divisor (GCD), which is the product of all common prime factors with the smallest powers they appear in both numbers. For 77 and 242, the GCD is 1111 because 11 is the only common prime factor, and it appears once in 77 and twice in 242 — so we take it once (the smaller power).
This method helps in simplifying ratios and fractions involving these numbers, determining divisibility, and even in cryptography and number theory.
To summarize, the common prime factor of 77 and 242 is 11. This simple process of prime factorization is a fundamental skill in mathematics that enables deeper understanding of number relationships.
