Number of Elements in U(n) for n = 248

To find the number of elements in U(n) for n = 248, we need to determine the set of positive integers less than 248 that are relatively prime to 248.

Definition: Relatively Prime
Two numbers are said to be relatively prime if their greatest common divisor (GCD) is 1. In other words, they do not share any common factors other than 1.

U(n) as a Group under Multiplication Modulo n
U(n) is a set of positive integers less than n that are relatively prime to n. We can show that U(n) forms a group under multiplication modulo n.

Group Properties:
1. Closure: For any two elements a, b in U(n), their product (a * b) modulo n is also in U(n).
2. Associativity: The operation of multiplication modulo n is associative.
3. Identity Element: The number 1 is an identity element in U(n) since 1 * a ≡ a * 1 ≡ a (mod n) for any a in U(n).
4. Inverse Element: For every element a in U(n), there exists an inverse element b in U(n) such that a * b ≡ b * a ≡ 1 (mod n).

Finding U(n) for n = 248:
To find U(n) for n = 248, we need to determine the positive integers less than 248 that are relatively prime to 248. We can use the following steps:

1. Prime Factorization of n:
248 = 2^3 * 31

2. Finding Numbers with Common Factors:
- Numbers divisible by 2: There are 248/2 = 124 numbers divisible by 2.
- Numbers divisible by 31: There are 248/31 = 8 numbers divisible by 31.
- Numbers divisible by both 2 and 31: There are 248/(2*31) = 4 numbers divisible by both 2 and 31.

3. Finding Numbers Relatively Prime to 248:
- Total Numbers: We have a total of 248 positive integers less than 248.
- Numbers with Common Factors: There are 124 + 8 - 4 = 128 numbers that have common factors with 248.
- Numbers Relatively Prime: The number of elements in U(248) = Total Numbers - Numbers with Common Factors = 248 - 128 = 120.

Conclusion:
Therefore, the number of elements in U(n) for n = 248 is 120. These elements are the positive integers less than 248 that are relatively prime to 248, meaning they do not share any common factors other than 1 with 248.