Investigation of a Number 420.

                                              Investigation of a Number 420:

If we look at a Number, we can think a lot of questions about that number. If a Number is interesting we can immediately note its properties. But even if we look at a seemingly uninteresting number we can try to analyze it deeply for better understanding of Number Theory. 

For example let us take a number N= 420. Just an ordinary Number from Mathematical point of view.

( In this discussion , Four Twenty has nothing to do with the Section 420 or the Waldo's Cannabis Culture!! This  has also nothing to do with the Angel Numbers or Numerology!)

There may be a lot of questions about this number or any other Number. Try to find answers of these questions for only N =420. It requires only elementary knowledge of Numbers, Progressions and Permutations/Combination and some persistence to answer these questions.

(1)    No of factors of N.

(2)    No. of odd factors of N.

(3)    No. of even factors of N.

(4)    Sum of all factors of N.

(5)    Sum of odd factors of N

(6)    Sum of even factors of N

(7)    Sum of reciprocal of all factors of N

(8)    Sum of reciprocals of even factors

(9)    Sum of reciprocals of odd factors

(10)Sum of squares of all factors of N

(11)Sum of cubes of all factors of N

(12)How many factors of N are multiple of 15

(13)How many factors of N are multiple of 20

(14)Product of all factors of N

(15)No of ways to write N as a product of two factors

(16)No of ways to write N as a product of three factors

(17)No of ways to write N as a product of four factors

(18)No of ways to write N as a product of five factors

(19)No. of ways to write N as a product of two co primes.

(20)Find the no of co primes to N that are less than N.
(21)Find the no of co primes to N that are less than 2N.
(22)Find the no of co primes to N that are less than 3N.

(23)Find the sum of sum all co primes to N that are less than N
(24)Find the sum of sum all co primes to N that are less than 2N

(25)No. of ways to write N as a product of one even factor and one odd factor.

(26)No of ways to write N as a product of two even factors.

(27) In how many ways can N be written as a sum of several Consecutive Natural numbers?

(28) In how many ways can reciprocal of N be written as a sum of reciprocals of two Natural Numbers?

(29)In how many ways can it be written as a difference of two perfect squares?

(30)What number should be added to it to make it a Perfect Number?

(31)What number should be subtracted from it to make it a Perfect Number?

(32) What will be the maximum area of a right angle triangle whose smallest side is equal to N?

(33) In how many ways can it be written as a sum of two even numbers.
(34) In how many ways can it be written as a sum of two odd numbers.
(35) If three nos are in AP such that their sum is equal to N, then find the possible no of ordered triplets.
(36) Find the number of factors of N having two prime factors.
(37) If abc =N find the no of positive integral solutions for a,b,and c.
(38) If abc =N find the no of negative integral solutions for a,b,and c.
(39) How many AP's can be formed in which first term is 1 and the last term is N?
(40) How many zeroes are there at the end of the number N! ?
(41) What is the right most non zero digit of N!?
(42) Let f(x) be the Product of all the composite numbers less than x. What is the the number of consecutive zeroes at the end of f(N)?

(43) How many factors less than N are factors of N² but not of N?

(44) A composite number has a total of N factors. If the Number of prime factors of N is maximum possible; how many pairs of factors exist which are co-prime to each other and where the product is equal to the Number itself?

(45) There are N identical coins. All the coins have the same weight equal to N except one which has a different weight. Also given is a common balance which can take any no of coins in each of the two pans. It is known that the coin which has different weight is lighter. what is the maximum no of weighings required to be certain of identifying the lighter coin?

Now a few difficult questions:

(46)  If all natural numbers from 1 to N are considered than how many of these can be written as a difference of two perfect squares in exactly one way?
(47)  If all natural numbers from 1 to N are considered than how many of these can be written as a difference of two perfect squares in exactly two way?
(48)  If all natural numbers from 1 to N are considered than how many of these can be written as a difference of two perfect squares in exactly three ways?
(49)  If all natural numbers from 1 to N are considered than how many of these can be written as a difference of two perfect squares in exactly four ways?
(50)  If all natural numbers from 1 to N are considered than how many of these can be written as a difference of two perfect squares in at most three ways?
(51)  If all natural numbers from 1 to N are considered than how many of these can be written as a difference of two perfect squares in at least ways?
(52)  If all natural numbers from 1 to N are considered than how many of these can be written as a difference of two perfect squares in zero ways?

(For the above questions from 46 to 52 ,  first you may take  N = 100 and later on take N= 420.)