The largest natural number, which exactly divides the product of any four consecutive natural numbers, is

### Concept & Formula A fundamental property of consecutive integers is that the product of any $n$ consecutive natural numbers is always perfectly divisible by $n!$ (n factorial). $$ n! = n \times (n-1) \times \dots \times 1 $$ Since we are looking at the product of $4$ consecutive natural numbers, the product will always be divisible by $4!$. ### Step-by-Step Solution * **Step 1: Understand the factors.** Any sequence of 4 consecutive integers will always contain: * At least one multiple of 2. * At least one multiple of 3. * At least one multiple of 4. * **Step 2: Combine the guaranteed factors.** Because the sequence guarantees these multiples, their product must be divisible by the product of these unique prime factors and their highest powers present. * **Step 3: Apply the factorial rule.** * $4! = 4 \times 3 \times 2 \times 1 = 24$. * **Conclusion:** The product of any four consecutive natural numbers is always divisible by 24. ### Exam Strategy & Shortcut **The Boundary Test (Fastest):** To find the *largest* universal divisor, simply compute the product of the *smallest* valid set of numbers. The smallest four consecutive natural numbers are 1, 2, 3, and 4. Product $= 1 \times 2 \times 3 \times 4 = 24$. The largest number that can divide 24 is 24 itself. This immediately gives you the absolute maximum possible divisor for all sequences. ### Common Pitfall Students often pick larger sequences (like 3, 4, 5, 6) which gives a product of 360. They then see that 360 is divisible by 120 (option d) and mistakenly select it. You must always test the absolute smallest sequence to find the true baseline divisor. ### Final Answer Therefore, the correct answer is **24**.