how many integers between 1 and 149 inclusive are relatively prime to 30
do you know what it means to be relatively prime?
The prime factors of 30 are 2,3,5
So how many multiples of those numbers are there?
Subtract the total from 149.
To find the number of integers between 1 and 149 inclusive that are relatively prime to 30, we need to determine how many numbers in this range do not share any common factors with 30 (except for 1).
Step 1: Prime factorize 30
The prime factorization of 30 is 2 * 3 * 5.
Step 2: Exclude numbers divisible by 2, 3, or 5
To find the numbers relatively prime to 30, we need to remove the numbers divisible by 2, 3, or 5.
There are 74 numbers divisible by 2 (from 2 to 148), 49 numbers divisible by 3, and 29 numbers divisible by 5.
Step 3: Find the numbers divisible by both pairs of primes
Since the numbers divisible by 2, 3, or 5 are counted twice, we need to subtract the numbers divisible by both pairs of primes (2 and 3, 2 and 5, 3 and 5) to avoid double-counting.
There are 24 numbers divisible by both 2 and 3 (from 6 to 144), 14 numbers divisible by both 2 and 5, and 9 numbers divisible by both 3 and 5.
Step 4: Calculate the number of numbers relatively prime to 30
To find the numbers relatively prime to 30, we subtract the numbers excluded in step 2 and add back the numbers excluded in step 3.
Total numbers relatively prime to 30 = (149 - 74 - 49 - 29) + (24 + 14 + 9)
= (149 - 152) + 47
= 47
Therefore, there are 47 integers between 1 and 149 inclusive that are relatively prime to 30.
To find the number of integers between 1 and 149 inclusive that are relatively prime to 30, we can use the concept of Euler's totient function.
Euler's totient function, denoted by φ(n), gives the count of positive integers that are less than or equal to n and are relatively prime to n.
To calculate φ(n), we need to find the prime factorization of n. In this case, the prime factorization of 30 is 2 * 3 * 5.
Now, we can use the formula for φ(n):
- φ(n) = n * (1 - 1/p1) * (1 - 1/p2) * ... * (1 - 1/pk)
where p1, p2, ..., pk are the distinct prime factors of n.
Applying this formula to our case, where n = 30:
- φ(30) = 30 * (1 - 1/2) * (1 - 1/3) * (1 - 1/5)
= 30 * (1/2) * (2/3) * (4/5)
= 8
Therefore, there are 8 integers between 1 and 149 inclusive that are relatively prime to 30.