Prime Factorization and the Number and Sum of Divisors
Prime factorization is a basic tool for investigating the properties of integers.
Prime Factorization
Expressing a natural number as a product of prime numbers. Example: 360 = 2³ × 3² × 5
Number of Divisors
When n = pa × qb × rc × …
Number of divisors = (a+1)(b+1)(c+1)…
Example ① (Number of Divisors)
Find the number of divisors of 360 = 2³ × 3² × 5¹.
Solution:(3+1)(2+1)(1+1) = 4×3×2 = 24 divisors
Find the number of divisors of 360 = 2³ × 3² × 5¹.
Solution:(3+1)(2+1)(1+1) = 4×3×2 = 24 divisors
Sum of Divisors
When n = pa × qb
Sum of divisors = (1+p+p²+…+pa)(1+q+q²+…+qb)
Example ② (Sum of Divisors)
Find the sum of the divisors of 12 = 2² × 3.
(1+2+4)(1+3) = 7×4 = 28
Find the sum of the divisors of 12 = 2² × 3.
(1+2+4)(1+3) = 7×4 = 28
Key Points
- An odd number of divisors ⟺ n is a perfect square (examples: 1,4,9,16,…)
- Prime factorization is performed by repeated division (try small primes in order)
- The sum of divisors is the product of the sums of geometric sequences for each prime factor
Practice Problems
- Prime-factorize 180 and find the number of divisors.
- Find the smallest natural number with exactly 6 divisors.
- Find the sum of the divisors of 72.
Solutions and Explanations
- 180 = 2²×3²×5. Number of divisors = (2+1)(2+1)(1+1) = 18 divisors
- Six divisors → if (5+1)=6, then p⁵ (minimum:2⁵=32), or if (2+1)(1+1)=6, then p²×q (minimum:2²×3=12) → 12
- 72=2³×3². (1+2+4+8)(1+3+9)=15×13=195