diff --git a/project_euler/problem_072/sol2.py b/project_euler/problem_072/sol2.py
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+"""
+Project Euler Problem 72: https://projecteuler.net/problem=72
+
+Consider the fraction, n/d, where n and d are positive integers. If n<d and HCF(n,d)=1,
+it is called a reduced proper fraction.
+
+If we list the set of reduced proper fractions for d ≤ 8 in ascending order of size,
+we get:
+
+1/8, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2,
+4/7, 3/5, 5/8, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 7/8
+
+It can be seen that there are 21 elements in this set.
+
+How many elements would be contained in the set of reduced proper fractions
+for d ≤ 1,000,000?
+"""
+
+
+def solution(limit: int = 1000000) -> int:
+    """
+    Return the number of reduced proper fractions with denominator less than limit.
+    >>> solution(8)
+    21
+    >>> solution(1000)
+    304191
+    """
+    primes = set(range(3, limit, 2))
+    primes.add(2)
+    for p in range(3, limit, 2):
+        if p not in primes:
+            continue
+        primes.difference_update(set(range(p * p, limit, p)))
+
+    phi = [float(n) for n in range(limit + 1)]
+
+    for p in primes:
+        for n in range(p, limit + 1, p):
+            phi[n] *= 1 - 1 / p
+
+    return int(sum(phi[2:]))
+
+
+if __name__ == "__main__":
+    print(f"{solution() = }")