Question

How many positive integers are there less than or equal to 150 which are relatively prime to 150 ?

• A. 40
• B. 42
• C. 38
• D. 44

Answer 1

Answer:

A. 40

There are 40 positive integers less than or equal to 150 that are relatively prime to 150.

To find how many numbers are relatively prime to a given integer (meaning they share no common factors other than 1), we use Euler's Totient Function, $\phi(n)$.

Step 1: Find the prime factorization of 150

• $150 = 2 \times 3 \times 5^2$

• The distinct prime factors ($p$) are 2, 3, and 5.

Step 2: Apply Euler's Totient formula
$\phi(n) = n \times \left(1 - \frac{1}{p_1}\right) \times \left(1 - \frac{1}{p_2}\right) \times \left(1 - \frac{1}{p_3}\right)$

Substitute the numbers into the formula:
$\phi(150) = 150 \times \left(1 - \frac{1}{2}\right) \times \left(1 - \frac{1}{3}\right) \times \left(1 - \frac{1}{5}\right)$

$\phi(150) = 150 \times \frac{1}{2} \times \frac{2}{3} \times \frac{4}{5}$

Step 3: Calculate the final value
$\phi(150) = 150 \times \frac{8}{30}$

$\phi(150) = 5 \times 8 = \mathbf{40}$