Counting Divisors

Number Theory · Axiom Academy

Discover surprising patterns in how many divisors numbers have. Let's start by exploring the number 12. Click on each number below to see if it divides evenly into 12. Which numbers divide evenly into 12? Now let's explore other numbers. Click on different numbers to see how many divisors each one has. Click to explore different numbers Step 3: The Perfect Square Mystery Use the slider to explore different numbers. Pay special attention to perfect squares! Let's investigate what happens when we look at powers of prime numbers. Divisors naturally come in pairs that multiply to give the original number. This is why most numbers have an even number of divisors. Perfect squares have an odd number of divisors because one divisor (the square root) pairs with itself rather than a different number. For prime powers p k , the divisor count follows a simple pattern: d(p k ) = k + 1. The divisors are exactly the powers from p⁰ to p k . For any number with prime factorization n = p₁ a₁ × p₂ a₂ × ... × p k a k , the divisor count is d(n) = (a₁ + 1)(a₂ + 1)...(a k + 1).

This is the written version of the interactive lesson above. See the full Number Theory course.