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Let U(n) be the set of all positive integers less than n and relatively prime to n for n = 248, the number of elements in U(n) is
- a)60
- b)120
- c)180
- d)240
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
Let U(n) be the set of all positive integers less than n and relativel...
A monoid(B,*) is called Group if to each element there exists an element c such that (a*c)=(c*a)=e. Here e is called an identity element and c is defined as the inverse of the corresponding element.
Most Upvoted Answer
Let U(n) be the set of all positive integers less than n and relativel...
Basically by definition of U(n), it's order is phi(n) where n is the no of elements in group. so phi (248) =248×(1-1/2) ×(1-1/31) =120 [ U should know about a little bit of euler's phi function and it's properties to compute this]
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Community Answer
Let U(n) be the set of all positive integers less than n and relativel...
Calculation of U(n):
To find the number of elements in U(n), we need to determine the integers less than 248 that are relatively prime to 248.
Euler's Totient Function:
Euler's Totient function, denoted by φ(n), gives the count of positive integers less than n that are relatively prime to n. It is defined as the number of positive integers less than n that are coprime to n.
Prime Factorization of n:
To calculate φ(n), we first need to find the prime factorization of n. For n = 248, the prime factorization is:
248 = 2^3 * 31
Calculating φ(n):
The formula to calculate φ(n) for a number with prime factorization p1^a1 * p2^a2 * ... * pk^ak is given by:
φ(n) = n * (1 - 1/p1) * (1 - 1/p2) * ... * (1 - 1/pk)
Applying this formula to n = 248, we get:
φ(248) = 248 * (1 - 1/2) * (1 - 1/31)
= 248 * (1/2) * (30/31)
= 120
Number of Elements in U(n):
The number of elements in U(n) is equal to φ(n), which is 120 in this case.
Therefore, the correct answer is option B) 120.
To find the number of elements in U(n), we need to determine the integers less than 248 that are relatively prime to 248.
Euler's Totient Function:
Euler's Totient function, denoted by φ(n), gives the count of positive integers less than n that are relatively prime to n. It is defined as the number of positive integers less than n that are coprime to n.
Prime Factorization of n:
To calculate φ(n), we first need to find the prime factorization of n. For n = 248, the prime factorization is:
248 = 2^3 * 31
Calculating φ(n):
The formula to calculate φ(n) for a number with prime factorization p1^a1 * p2^a2 * ... * pk^ak is given by:
φ(n) = n * (1 - 1/p1) * (1 - 1/p2) * ... * (1 - 1/pk)
Applying this formula to n = 248, we get:
φ(248) = 248 * (1 - 1/2) * (1 - 1/31)
= 248 * (1/2) * (30/31)
= 120
Number of Elements in U(n):
The number of elements in U(n) is equal to φ(n), which is 120 in this case.
Therefore, the correct answer is option B) 120.
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