東京大学情報理工学系研究科創造情報学専攻 2007年8月実施筆記試験第1問

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Let be the number of divisors of a positive integer . Let us compute the smallest for a given . Note that and are included among the divisors of .

(1) Calculate the smallest each for and .

(2) Let be prime factorized as where s are mutually different prime numbers and s are positive integers for . Describe in a mathematical formula.

(3) When is odd, what kind of number is ?

(4) Based on (2), describe the outline of a method to compute the smallest given . Moreover, describe ways to decrease computational complexity.

(5) Calculate the smallest for .