Prime factorization of n!

Rather than the conventional ways of prime factorization, there is an easy way to do the prime factorization of n!. First of all you have to list out all the prime numbers less than or equal to n. It is because 

n! = 1 x 2 x 3 x 4 x 5 x .... x n 

So prime factors of n! are all the prime numbers less than n. Next we have to find out the multiplicity of each of these prime factors. 

Google doodle for Srinivasa Ramanujan's 125th birthday

Srinivasa Ramanujan's 125th birthday marked by Google India doodle on 22 December 2012.

Locker puzzle

1. A high school has a strange principal. On the first day, he has his students perform an odd opening day ceremony:
There are one thousand lockers and one thousand students in the school. The principal asks the first student to go to every locker and open it. Then he has the second student go to every second locker and close it. The third goes to every third locker and, if it is closed, he opens it, and if it is open, he closes it. The fourth student does this to every fourth locker, and so on. After the process is completed with the thousandth student, how many lockers are open?

Number Patterns

Some amazing number patterns:

        1 x 1                 =                         1

              11 x 11               =                      1 2 1

            111 x 111             =                   1 2 3 2 1

          1111 x 1111           =                1 2 3 4 3 2 1

        11111 x 11111         =             1 2 3 4 5 4 3 2 1

      111111 x 111111       =          1 2 3 4 5 6 5 4 3 2 1

    1111111 x 1111111     =       1 2 3 4 5 6 7 6 5 4 3 2 1

  11111111 x 11111111   =    1 2 3 4 5 6 7 8 7 6 5 4 3 2 1

111111111 x 111111111 = 1 2 3 4 5 6 7 8 9 8 7 6 5 4 3 2 1

Story of 1729 : The first taxicab number

1729 is known as the Hardy–Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a hospital visit to the Indian mathematician Srinivasa Ramanujan. In Hardy's words: 

I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."

G. H. Hardy and Srinivasa Ramanujan 

The two different ways are these:

1729 = 1³ + 12³ = 9³ + 10³

Generalizations of this idea have created the notion of "taxicab numbers" ( The nth taxicab number Ta(n) is the smallest number representable in n ways as a sum of positive cubes).
The number was also found in one of Ramanujan's notebooks dated years before the incident, and was noted by Frénicle de Bessyin 1657.

The Prince Of Mathematics

Carl Friedrich Gauss

In 1787 at a german school, the schoolmaster gave the following assignment to his students: "Write down all the whole numbers from 1 to 100 and add up their sum". After finishing the assignment, each student has to bring his slate forward and place it on the schoolmaster's desk. The teacher expected the beginner's class to take a good while to finish this exercise. But in a few seconds, to his teacher's surprise, a student proceeded to the front of the room and placed his slate on the desk. The schoolmaster was astounded to see only one number: 5050 in that student's slate. That boy then had to explain to his teacher that he found the result because he could see that, 1+100=101, 2+99=101, 3+98=101, so that he could find 50 pairs of numbers that each add up to 101. Thus, 50 times 101 will equal 5,050.