Fix the mistake: Properties of Integers ① (Divisors and Multiples)

Prime Factorization and the Number and Sum of Divisors

Prime factorization is a basic tool for investigating the properties of integers.

Expressing a natural number as a product of prime numbers. Example: 360 = 2³ × 3² × 5

When n = p^a × q^b × r^c × …
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

When n = p^a × q^b
Sum of divisors = (1+p+p²+…+p^a)(1+q+q²+…+q^b)

Example ② (Sum of Divisors)
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

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