Let U(n) be the set of all positive integers less than n and relatively prime to n for n = 248, the numberof elements in U(n) isa)60b)120c)180d)240Correct answer is option 'B'. Can you explain this answer?

Question

Answer

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]

Answer

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.

Answer

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.

Answer

We can use Euler function formula too
no. of generators=( p^n- p^n-1)
order of group=248= 2^3*31
no. of generators= (2³-2²)(31¹-31⁰)=(8-4)(31-1)=(4)(30)
=4*30=120

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