diff --git a/src/main/java/com/thealgorithms/maths/Prime/MobiusFunction.java b/src/main/java/com/thealgorithms/maths/Prime/MobiusFunction.java
index 3d4e4eff0f03..ec1785a916c7 100644
--- a/src/main/java/com/thealgorithms/maths/Prime/MobiusFunction.java
+++ b/src/main/java/com/thealgorithms/maths/Prime/MobiusFunction.java
@@ -31,27 +31,28 @@ public static int mobius(int number) {
             throw new IllegalArgumentException("Number must be greater than zero.");
         }
 
-        if (number == 1) {
-            // return 1 if number passed is less or is 1
-            return 1;
-        }
-
         int primeFactorCount = 0;
+        int remaining = number;
 
-        for (int i = 1; i <= number; i++) {
-            // find prime factors of number
-            if (number % i == 0 && PrimeCheck.isPrime(i)) {
-                // check if number is divisible by square of prime factor
-                if (number % (i * i) == 0) {
-                    // if number is divisible by square of prime factor
+        /* Divide out every prime factor in turn. Trial division only has to run up to the square
+        root of the remaining value, and the multiplication is widened to long so that the bound
+        does not overflow for numbers close to Integer.MAX_VALUE. */
+        for (int factor = 2; (long) factor * factor <= remaining; factor++) {
+            if (remaining % factor == 0) {
+                remaining /= factor;
+                if (remaining % factor == 0) {
+                    // number is divisible by the square of this prime factor
                     return 0;
                 }
-                /*increment primeFactorCount by 1
-                                if number is not divisible by square of found prime factor*/
                 primeFactorCount++;
             }
         }
 
+        /* Whatever is left is either 1 or a single prime factor larger than the square root. */
+        if (remaining > 1) {
+            primeFactorCount++;
+        }
+
         return (primeFactorCount % 2 == 0) ? 1 : -1;
     }
 }
diff --git a/src/test/java/com/thealgorithms/maths/prime/MobiusFunctionTest.java b/src/test/java/com/thealgorithms/maths/prime/MobiusFunctionTest.java
index 734d02477ba2..98ac29e81a5b 100644
--- a/src/test/java/com/thealgorithms/maths/prime/MobiusFunctionTest.java
+++ b/src/test/java/com/thealgorithms/maths/prime/MobiusFunctionTest.java
@@ -5,6 +5,8 @@
 
 import com.thealgorithms.maths.Prime.MobiusFunction;
 import org.junit.jupiter.api.Test;
+import org.junit.jupiter.params.ParameterizedTest;
+import org.junit.jupiter.params.provider.CsvSource;
 
 class MobiusFunctionTest {
 
@@ -152,4 +154,15 @@ void testMobiusFunction() {
             assertEquals(expectedValue, actualValue);
         }
     }
+
+    /**
+     * Large inputs whose smallest square divisor test used to overflow, most notably
+     * {@code Integer.MAX_VALUE}, whose square wraps around to 1 and made every number look like it
+     * had a squared prime factor.
+     */
+    @ParameterizedTest
+    @CsvSource({"2147483647, -1", "2147483646, 0", "2147483645, -1", "2147483644, 0", "2147483629, -1", "2147395600, 0", "1073741824, 0", "1073741789, -1", "999999937, -1", "999999999, 0", "2146689000, 0"})
+    void testMobiusForLargeNumbers(int number, int expected) {
+        assertEquals(expected, MobiusFunction.mobius(number));
+    }
 }