Find the number of co-primes to the number 144, that are less than the number itself?*1 point48326456

To find the number of co-primes to the number 144 that are less than the number itself, we need to find the Euler's totient function of 144.

The Euler's totient function, φ(n), is used to determine the number of positive integers that are co-prime to n.

Step 1: Prime factorize the number 144. The prime factors of 144 are 2 and 3.

Step 2: Use the formula for Euler's totient function, which is n(1 - 1/p1)(1 - 1/p2)...(1 - 1/pk), where p1, p2, ..., pk are the distinct prime numbers dividing n.

Step 3: Substitute the values into the formula. So, φ(144) = 144(1 - 1/2)(1 - 1/3) = 144 * 1/2 * 2/3 = 48.

So, there are 48 numbers less than 144 that are co-prime to 144.

To find the number of co-primes to the number 144 that are less than the number itself, we first need to understand what co-primes are. Co-primes are numbers whose greatest common divisor (GCD) is 1. In other words, two numbers are co-primes if the only positive integer that divides both of them is 1.

Now, let's find the co-primes of 144.

Step 1: Find the prime factorization of 144. The prime factorization of 144 is 2^4 * 3^2.

Step 2: Use Euler's Totient function to find the number of co-primes. The formula for Euler's Totient function is φ(n) = n * (1 - 1/p1) * (1 - 1/p2) * ... * (1 - 1/pk), where p1, p2, ..., pk are the distinct prime numbers that divide n.

Step 3: Apply the formula to 144. φ(144) = 144 * (1 - 1/2) * (1 - 1/3) = 144 * 1/2 * 2/3 = 48.

So, there are 48 numbers less than 144 that are co-prime to 144.

To find the number of co-primes to the number 144 that are less than the number itself, we need to find the Euler's totient function of 144.

The Euler's totient function φ(n) is an arithmetic function that counts the positive integers less than a given integer n that are relatively prime to n.

The formula for Euler's totient function when n is a product of two distinct prime numbers is given by:

φ(n) = n * (1 - 1/p) * (1 - 1/q)

Where p and q are prime factors of n.

The prime factorization of 144 = 2^4 * 3^2

So, φ(144) = 144 * (1 - 1/2) * (1 - 1/3) = 144 * 1/2 * 2/3 = 48

So, there are 48 numbers less than 144 that are co-prime to 144.