Question

The number of positive integers less than or equal to n that are co-prime to n is called the _____________ of n.

So for the number 6, 4 and 5 are co-prime with 6. Therefore, ______________ = 2. A) Prime Factorization

B) Euler's Totient Function

C) Modular Arithmetic

D) Fermat's Little Theorem

Answer 1

Step-by-step explanation:

The correct answer is:

B) Euler's Totient Function

The number of positive integers less than or equal to n that are co-prime to n is called Euler's Totient Function, often denoted as φ(n). Euler's Totient Function calculates the count of numbers that are co-prime to a given number.

In the example provided, the number is 6. To find the count of numbers that are co-prime to 6, we need to determine the positive integers less than or equal to 6 that do not share any common factors other than 1 with 6. These co-prime numbers are 1, 5, 7, 11, 13, and so on.

In this case, the numbers 4 and 5 are co-prime with 6. Therefore, the value of φ(6) (Euler's Totient Function of 6) is 2.

Here is a breakdown of the other answer choices:

A) Prime Factorization: Prime factorization is a method used to express a number as the product of its prime factors. While prime factorization is related to finding prime numbers, it does not directly refer to the count of numbers that are co-prime to a given number.

C) Modular Arithmetic: Modular arithmetic is a system of arithmetic for integers where numbers "wrap around" after reaching a certain value called the modulus. While modular arithmetic is a useful concept, it does not specifically refer to the count of co-prime numbers.

D) Fermat's Little Theorem: Fermat's Little Theorem is a theorem in number theory that establishes a relationship between prime numbers and modular arithmetic. While it is a relevant concept, it does not directly refer to the count of numbers that are co-prime to a given number.

Therefore, the correct answer is B) Euler's Totient Function, as it accurately describes the concept of counting co-prime numbers, such as the numbers co-prime to 6 in the given example.

Answer 2

The number of positive integers less than or equal to n that are co-prime to n is called Euler's Totient Function. To find the number of positive integers that are co-prime to n, we need to factorize n into its prime factors.

Step-by-step explanation:

The number of positive integers less than or equal to n that are co-prime to n is called Euler's Totient Function.

To find the number of positive integers that are co-prime to n, we need to factorize n into its prime factors. For example, if n = 6, the prime factors are 2 and 3. We can use the formula φ(n) = n * (1 - 1/p₁) * (1 - 1/p₂) * ... * (1 - 1/pₙ), where p₁, p₂, ..., pₙ are the prime factors of n. For n = 6, φ(n) = 6 * (1 - 1/2) * (1 - 1/3) = 6 * (1 - 0.5) * (1 - 0.33) = 2.