Introduce my masterpieces of math problem

I sometimes make problems and post them in Japanese local contest yukicoder. I want many people to know math problems I made, so I introduce some of them. Let's try!!

LCMST

Problem Statement

Consider a complete graph on vertices. The weight of edge between vertex and is least common multiple of and . Find the total weight of the edges contained in a minimum spanning tree of .

Constraints

• Time Limit is 4000ms

Sample

Consider the case where and . The answer is . This is optimal to draw edges between . These have weights , respectively.

Hints

Simple Math ?

Problem Statement

You are given integer . Find . You have test cases to solve.

Constraints

• Time Limit is 2000ms

Sample

Consider the case where and . The answer is .

Hints

Simple Math !

Problem Statement

You are given integers . Find the -th smallest positive integer that satisfies the following conditions:

• There exists a pair of nonnegative integers such that

You have test cases to solve.

Constraints

• Time Limit is 2000ms

Sample

Consider the case where . The answer is .

• When , there is no pair of nonnegative integers satisfying .
• When , there is no pair of nonnegative integers satisfying .
• When , satisfies .
• When , satisfies .
• When , there is no pair of nonnegative integers satisfying .
• When , satisfies .
• When , satisfies .

Hints