A particular fraction has a prime number in the numerator and a different prime number in
the denominator. Justify whether or not this fraction can be simplified and give reasons.
What is the definition of a prime number?
https://whatis.techtarget.com/definition/prime-number
What do you think?
If a particular fraction has a prime number in the numerator and a different prime number in the denominator, then the fraction cannot be simplified any further.
This is because a prime number has no divisors other than 1 and itself. Since both the numerator and denominator are prime numbers, there are no common factors between them. As a result, there is no way to simplify the fraction further by canceling out any common factors.
For example, let's consider the fraction 3/5. Both 3 and 5 are prime numbers, and they have no common factors other than 1. Therefore, the fraction 3/5 is already in its simplest form and cannot be further simplified.
Hence, if a fraction has a prime number in the numerator and a different prime number in the denominator, it cannot be simplified any further.
To determine whether or not a fraction with a prime number in the numerator and a different prime number in the denominator can be simplified, we need to consider the properties of prime numbers and fractions.
First, let's establish some facts:
1. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself (e.g., 2, 3, 5, 7, 11, etc.)
2. A fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD).
3. If two prime numbers are different, their GCD is 1 because there are no common factors other than 1.
Considering these facts, it follows that a fraction with a prime number in the numerator and a different prime number in the denominator cannot be simplified. This is because the GCD of the numerator and denominator is already 1, and dividing both by 1 does not change the fraction.
For example, let's consider the fraction 3/7. Both 3 and 7 are prime numbers, and their GCD is 1. Dividing both the numerator and the denominator by their GCD (which is 1) yields the fraction 3/7 again. Hence, the fraction cannot be simplified further.
In general, if a fraction has a prime number in the numerator and a different prime number in the denominator, it cannot be simplified because their GCD is 1.