Pascal's Triangle

　I often solve some problems in the intervals between my study. Today, I chose a problem with Pascal's triangle from Project Euler. Pascal's triangle is a triangle that is created by the simple rules, such as following:

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1

　And the problem demanded the number of entries which are not divisible by 7 in the first one billion rows of Pascal's triangle. The number of elements that I have to check is about 500 quadrillion (5 times 10 to the 17th power).

　I wrote a simple program code which calculates Pascal's triangle made of residue of 7 (modularized by 7?). However, this program took about 30 seconds to check 5 billion elements in my computer. This means it will take 95 years to solve this problem. I think there is a smart analytical solution because there are too many elements to solve using dynamic programming. I couldn't solve this problem today, but I like such a problem that remind me of the splendor of algorithms.

• I often solve some problems in the intervals between my study.
• I often solve some problems (or: mathematical puzzles) in the intervals between my study (or: studies).

• I couldn't solve this problem today, but I like such a problem that remind me of the splendor of algorithms.
• I couldn't solve this problem today, but I like such problems that remind me of the splendor of algorithms.

Interesting!

• Today, I chose a problem with Pascal's triangle from Project Euler.
• This sentence is perfect! No correction needed!

• Pascal's triangle is a triangle that is created by the simple rules, such as following:
• Pascal's triangle is a triangle that is created by the simple rules, such as the following:

• And the problem demanded the number of entries which are not divisible by 7 in the first one billion rows of Pascal's triangle.
• And the problem demanded the number of entries which are not divisible by 7 in the first one billion rows of Pascal's triangle.

  You can also use "asked for" instead of "demanded" here

• The number of elements that I have to check is about 500 quadrillion (5 times 10 to the 17th power).
• This sentence is perfect! No correction needed!

• I wrote a simple program code which calculates Pascal's triangle made of residue of 7 (modularized by 7?).
• I wrote a simple program code which calculates Pascal's triangle made of residue of 7 (modularized by 7?).

  I would use "modularized by 7" or just "modulo 7".

• However, this program took about 30 seconds to check 5 billion elements in my computer.
• This sentence is perfect! No correction needed!

• This means it will take 95 years to solve this problem.
• This sentence is perfect! No correction needed!

• I think there is a smart analytical solution because there are too many elements to solve using dynamic programming.
• This sentence is perfect! No correction needed!

This was very interesting and very well written. I wish I had a better idea how to solve this problem.