Solution: We are to count how many integers from 1 to 120 are relatively prime to 12. This is given by Euler’s totient function applied to the set of positions, but since the condition is relative primality with 12, we compute $ \phi(12) $, the number of integers from 1 to 12 that are coprime to 12, and then multiply by the number of full cycles in 120.

["Counting Integers Relatively Prime to 12: A Complete Guide Using Euler’s Totient Function", "When solving number theory problems involving relative primality, one essential tool is Euler’s totient function, denoted $ \phi(n) $. It counts how many integers from 1 to $ n $ are coprime (relatively prime) to $ n $. In this article, we explore how to compute the number of integers from 1 to 120 that are relatively prime to 12 by applying Euler’s totient function.", "### What Does It Mean for an Integer to Be Relatively Prime to 12?", "Two integers are relatively prime if their greatest common divisor (GCD) is 1. For example, integers that share no prime factors with 12 — which factors into $ 2^2 \ imes 3 $ — qualify. Therefore, an integer $ k $ from 1 to 120 is relatively prime to 12 if $ \gcd(k, 12) = 1 $.", "### Step 1: Compute $ \phi(12) $ — The Count for One Cycle", "Euler’s totient function for a number based on its prime factorization is:", "$$\n\phi(n) = n \left(1 - \frac{1}{p_1}\right)\left(1 - \frac{1}{p_2}\right)\cdots\left(1 - \frac{1}{p_k}\right)\n$$", "since $ 12 = 2^2 \ imes 3 $, and we only consider distinct primes $ {2, 3} $.", "So,", "$$\n\phi(12) = 12 \left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{3}\right) = 12 \ imes \frac{1}{2} \ imes \frac{2}{3} = 12 \ imes 0.5 \ imes 0.666... = 12 \ imes \frac{1}{3} = 4\n$$", "Thus, there are 4 integers from 1 to 12 that are relatively prime to 12.", "These integers are: $ 1, 5, 7, 11 $. Checking each, we confirm $ \gcd(k, 12) = 1 $ for each.", "### Step 2: Determine How Many Full Cycles Fit in 1 to 120", "Since $ 120 \div 12 = 10 $, there are 10 complete cycles of 1 to 12 within the range 1 to 120.", "### Step 3: Multiply to Find Total Relatively Prime Integers", "Because relative primality repeats every 12 numbers, each block of 12 contributes $ \phi(12) = 4 $ integers coprime to 12.", "Therefore, total integers relatively prime to 12 from 1 to 120 is:", "$$\n10 \ imes \phi(12) = 10 \ imes 4 = 40\n$$", "Alternatively, by explicit computation: list all numbers from 1 to 12 coprime to 12 (4 numbers), then repeat the pattern 10 times — again yielding 40.", "### Final Answer", "There are 40 integers from 1 to 120 that are relatively prime to 12, computed via Euler’s totient function applied over repeated cycles.", "---", "Summary:", "- $ \phi(12) = 4 $\n- Range: 1 to 120 contains 10 full cycles of 12\n- Total relatively prime integers: $ 10 \ imes 4 = 40 $", "This elegant application of Euler’s totient function allows efficient counting in modular arithmetic and cryptographic algorithms where coprimality is essential.", "---", "Keywords: Euler’s totient function, relatively prime count, $ \phi(12) $, integers coprime to 12, 1 to 120, counting coprime numbers, number theory, GCD, 12 totient."]