SANITATION

Current population estimates indicate that there are 7.4 billion people in the World as of the year 2017. The World Health Organization (WHO) estimates that close to 950 million people in our World defecate outside…….with almost 600 million of these people living in India. In India the practice of social caste and religious customs cause some people to pass over public sanitation available to them in preference for taking care of matters outdoors. Many of the African and Southeast Asian countries and some of the Carribean Islands also have high percentages of people who defecate outdoors. The situation brings about the spread of human diseases including deadly hepatitis infections and deadly cholera infections. In India alone more than 100,000 children 5 years old or younger die each year due to diseases spread by open defecation…….with flies being the most important carrier of the diseases spread to humans. WHO estimates show that despite these shocking statistics the situation with sanitation in the World at large has been improving over the past few decades……..with more and more public sanitation facilities being built and functioning nowadays to give people a choice to be safer and cleaner.

PROOF

given the conjecture:

“the sequence of the last digits in the sequences of the powers of the integers reoccur every fourth power and repeat in 10 digit intervals within a given sequence”

a partial proof of this statement may be:

(1) any number abcd……..wxyz may be expressed as a sum (a(x)10^n + b(x)10^n-1 + …… + y(x)10^1 + z(x)10^0)

(2) given (1) any number ab may be expressed as the sum (a(x)10^1 + b(x)10^0)………..thus by the binomial theorem any number (ab)^n…….is such that (ab)^n = (a^n(x)10^1(x)n + a number of cross terms + b^n(x)10^0(x)n)

(3) it is clear thus from the binomial theorem in (2) that the last digit in any number (ab)^n is determined only by the last digit in the original number ab…….that is by “b” itself as multiplied successively in the product (ab)^n 

(4) and any term in the binomial expansion of a number (ab)^n is itself multiplied by a coefficient from Pascal’s triangle in which the leading term a^n and the trailing term b^n are multiplied by a coefficient of just “1”

(5) and thus the generalization of the binomial theorem of (a + b)^n is therefore allowable to a general summation (a + b +…..+ y + z)^n………and in the number (ab……yz)^n only the last digit in the original number “z” contributes to the last digit in the product