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Find all factors of a positive integer and display its complete prime factorization with exponents.
Finding Factors:
For n, check every integer d from 1 to sqrt(n). If n % d == 0, then d and n/d are both factors.
Prime factorization: repeatedly divide by the smallest prime factor.
Example: 360 = 2^3 x 3^2 x 5
A factor of a number n is any integer that divides n exactly with no remainder. For example, factors of 12 are 1, 2, 3, 4, 6, and 12.
Prime factorization expresses a number as a product of its prime factors. For example, 60 = 2² x 3 x 5. Every integer greater than 1 has a unique prime factorization (Fundamental Theorem of Arithmetic).
If n = p1^a1 x p2^a2 x ... x pk^ak, then the total number of factors is (a1+1)(a2+1)...(ak+1). For example, 12 = 2²x3, so it has (2+1)(1+1) = 6 factors.
A prime number has exactly 2 factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13. The number 1 is not prime by convention.
A perfect number equals the sum of all its factors excluding itself. For example, 6 is perfect because 1 + 2 + 3 = 6. The next perfect numbers are 28, 496, and 8128.