Tag Archives: factor

386. Antichain

Let n be an integer and S(n) be the set of factors of n.

A subset A of S(n) is called an antichain of S(n) if A contains only one element or if none of the elements of A divides any of the other elements of A.

For example: S(30) = {1, 2, 3, 5, 6, 10, 15, 30} , {2, 5, 6} is not an antichain of S(30).  {2, 3, 5} is an antichain of S(30).

Let N(n) be the maximum length of an antichain of S(n).  Find ΣN(n) for 1 ≤ n ≤ 108

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15. Grid Routes

Starting in the top left corner of a 2×2 grid, there are 6 routes (without backtracking) to the bottom right corner.

How many routes are there through a 20×20 grid? Read more of this post

12. Trigangle Numbers

The sequence of triangle numbers is generated by adding the natural numbers. So the 7th triangle number would be 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28. The first ten terms would be: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, … Let us list the factors of the first seven triangle numbers:

1 => 1: 1
2 => 3: 1,3
3 => 6: 1,2,3,6
4 => 10: 1,2,5,10
5 => 15: 1,3,5,15
6 => 21: 1,3,7,21
7 => 28: 1,2,4,7,14,28 = 6
8 => 36= 1,2,3,4,6,9,12,13,16,18 = 10
9 => 45 = 1,3,5,9,15,45 = 6
We can see that 28 is the first triangle number to have over five divisors. What is the value of the first triangle number to have over five hundred divisors? Read more of this post

3.Largest Prime Factor

The prime factors of 13195 are 5, 7, 13 and 29.  What is the largest prime factor of the number 600851475143 ?

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