HOW TO FIND THE LAST DIGIT OF ANY NUMBER

How to find the last digit of a number?

It is very difficult to find the last digit of a number by usual traditional method. In number theory finding the last digit or finding the remainder when any number is divided by 10 is having more importance and also one of interesting problem.

However if u remember and used the following rule, we can easily find the last digit.

Last digit of the number of the form mⁿ
m n=1 n=2 n=3 n=4 n=5
0 0 0 0 0 0
1 1 1 1 1 1
2 2 4 8 6 2
3 3 9 7 1 3
4 4 6 4 6 4
5 5 5 5 5 5
6 6 6 6 6 6
7 7 9 3 1 7
8 8 4 2 6 8
9 9 1 9 1 9

Last digit of mⁿ is changing upto n=4 when n=5, the last digit is repeated. Hence the last digit is periodic with 4

METHOD: To find the last digit of number mⁿ Divide n by 4, If the remainder is r, then the last digit of the number is (the last digit of given number)^r. otherwise if remainder is zero, then the last digit of number is last digit of m^4 i.e (last digit of given number)^4.

ILLUSTRATIONS:
1) The last digit of the number
N=(2003)⁵⁰⁰³ +(6007)⁹⁰⁵ –(209)⁶⁰⁸ =3³ +7¹ -9⁴ =14-1=13, the last digit is 3.
2)Find the remainder when

(608)¹⁸²⁹ is divided by 10

Sol: Finding the remainder when any number is divided by 10 is same as finding the last digit of the number , since when any number is divisible by 10 the remainder is the last digit of the given number. Last digit of 608 is 8 and when 1829 is divided by 4 , remainder is 1

Hence the last digit is 8¹ (by rule )
3)Find the last digit of 4^1328
Here when 1328 is divided by 4 , Remainder is 0 , so the last digit of 4^1328 is last digit of 4^4 i.e 6
FINDING THE LAST DIGIT ANY NUMBER OF THE FORM m^n ENDING WITH 0,1,2,3,4,5,6,7,8,9
a)The last digit of 5^1 is 5
5^2 is 5
5^3 is 5
Hence the last digit of 5^any number is 5
b)The last digit 6^1 is 6
6^2 is 6
6^3 is 6
and so on.
Hence the last digit of 6^any number is 6
c)The last digit of
11^1 is 1
11^2 is 1
11^3 is 1
11^4 is 1 and so on
The last digit of (any number ending with 1)^any number is 1
d)Similarly we can show that the last digit of (any number any number ending with 0)^any number is 0

e)Last digit of 2^1 is 2
2^2 is 4
2^3 is 8
2^4 is 16 ie 6
2^5 is 32 ie 2
2^6 is 64 i.e 4
2^7 is 128 i.e 8
2^8 is 256 i.e 6
In general we can see that the last digit of (any number ending with 2)^any number is 2,4,8,6
when power n is odd , the last digit is 2 and 8; when the power is even, the last digit is 4 and 6
2²ⁿ ≡ 6(mod 10) when n is even

≡ 4(mod 10) when n is odd

f)3²ⁿ ≡ 1(mod 10) when n is even .

≡9(mod 10) when n is odd

g) We can also see that last digit of (any number ending with 7)^any number is 1 when n is odd and 9 when n is even
7²ⁿ ≡1(mod 10) when n is even.

≡ 9(mod 10) when n is odd

h)The last digit of (any number ending with 8)^any number is is 8, 6 or 4

8²ⁿ ≡ 6(mod 10) when n is even.

≡ 4(mod 10) when n is odd.

j)9ⁿ ≡ 1(mod 10) when n is even

≡ 9(mod 10) when n is odd

To find the unit or last digit remember the following:(Try to prove)

a)5ⁿ≡5(mod 10) b)6ⁿ =6(mod 10)

c)4ⁿ ≡ 6(mod 10) when n is even .

≡ 4(mod 10) when n is odd

d)9ⁿ ≡ 1(mod 10) when n is even

≡ 9(mod 10) when n is odd

e)3²ⁿ ≡ 1(mod 10) when n is even .

≡ 9(mod 10) when n is odd

f)2²ⁿ ≡ 6(mod 10) when n is even

≡ 4(mod 10) when n is odd

g)8²ⁿ ≡ 6(mod 10) when n is even.

≡ 4(mod 10) when n is odd.

h) 7²ⁿ ≡1(mod 10) when n is even.

≡ 9(mod 10) when n is odd i.e

When n is positive integer ,

the last digit of 5ⁿ is 5 ; the last digit in 6ⁿ is 6.

The last digit in 3⁴ⁿ is 1; The last digit in 9ⁿ is 1 or 9 according as n is odd or even. The last digit of 11ⁿ is 1, 6ⁿ is 6 , 4ⁿ is 4 or 6 according as n is odd or even.

examples:

Unit digit of 17¹⁸⁹ =unit digit of 7¹⁸⁹ =7

Unit digit of 28²⁰⁰ =unit digit of 8²⁰⁰ =6

Unit digit of 43²⁰ =unit digit of 3²⁰ =1

Unit digit of 79²⁷ =unit digit of 9²⁷ =9

Unit digit of 84³⁷ =unit digit of 4³⁷ =4

STORY OF ZERO

STORY OF ZERO

Perhaps the most fundamental contribution of ancient India to the progress of civilisation is the decimal system of numeration including the invention of the number zero. This system uses 9 digits and a symbol for zero to denote all integral numbers, by assigning a place value to the digits. This system was used in Vedas and Valmiki Ramayana. Mohanjodaro and Harappa civilisations (3000 B.C.) also used this system.

If zero merely signified a magniutude or a direction seperatorI(i.e. separting those above the zero level from those below the zero level), the Egyptian zero, nfr, dating back atleast four thousand years, amply served these purposes. The ancient Egyptians (5000 B.C.) had a system based on 10, but they didn't use positional notation. Thus to represent 673, they would draw six snares, seven heel bones and three vertical strokes.

If zero was merely a place holder symbol, indicating the absence of a magnitude at a specified place position (such as, for example, the zero in 10 indicated the absence of any 'tens' in one hundred and one), then such a zero was already present in the babylonian number system long before the first recorded occurence of the Indian zero. Babylonians in Mesopotamia (3000 B.C.) had a sexagesimal system using base 60. Greeks and Romans had a cumbersome system (try to write 2376 in Roman numerals).

If zero was represented by just an empty space within a well defined postional number system, such zero was present chinese mathematics a few centuries before the Indian zero.

Many civilisations had some concept of "zero" as nothing - for example, if you have two cows and they both die, you are left with nothing.

However, the Indians were the first to see that zero can be used for something beyond nothing - at different places in a number, it adds different values. For example, 76 is different from 706, 7006, 760 etc.

Indian zero alluded to in the question was a multi faceted mathematical object: a symbol, a number, a magnitude, a direction seprator and a place holder, all in one operating with a fully established positional number system. Such a zero occured only twice in history- the indian zero which is now the universal zero and the Mayan zero which occupied in solitary isolation in central america around the beginning of commaon Era.

Brahmgupta (598 AD - 660 AD) was the first to give the rules of operation of zero.

A + 0 = A, where A is any quantity.
A - 0 = A,
A x 0 = 0,
A / 0 = 0

He was wrong regarding the last formula. This mistake was corrected by Bhaskara (1114 AD - 1185 AD), who in his famous book Leelavati, claimed that division of a quantity by zero is an infinite quantity or immutable God.

The ancient Indians represented zero as a circle with a dot inside. The word 'zero' comes from the Arabic "al-sifr". Sifr in turn is a transiliteration of the Sanskrit word "soonya" meaning of void or empty, which became later the term for zero. This and the decimal number system fascinated Arab scholars who came to India. Arab mathematician Al-Khowarizmi (790 AD - 850 AD) wrote Hisab-al-Jabr wa-al-Muqabala (Calculation of Integration and Equation) which made Indian numbers popular. "Soonya" became "al-sifr" or "sifr". The impact of this book can be judged by the fact that "al-jabr" became "Algebra" of today.

An Italian Leonardo Fibonacci (1170 AD - 1230 AD) took this number system to Europe. The Arabic "sifr" was called "zephirum" in Latin, and acquired many local names in Europe including "cypher". In the beginning, the merchants used to Roman numbers found the decimal system a new idea, and referred to these numbers as "infidel numbers", as the Arabs were called infidels because they had invaded the holy land of Palestine.

However, nowadays this system is called Hindu-Arabic System. This positional system of representing integers revolutionised the mathematical calculations and also helped in Astronomy and accurate navigation. The use of positional system to indicate fractions was introduced around 1579 AD by Francois Viete. The dot for a decimal point came to be used a few years later, but did not become popular until its use by Napier.

CALCULUS JOKES

• A mathematician went insane and believed that he was the differentiation operator. His friends had him placed in a mental hospital until he got better. All day he would go around frightening the other patients by staring at them and saying "I differentiate you!"

One day he met a new patient; and true to form he stared at him and said "I differentiate you!", but for once, his victim's expression didn't change.

Surprised, the mathematician marshalled his energies, stared fiercely at the new patient and said loudly "I differentiate you!", but still the other man had no reaction. Finally, in frustration, the mathematician screamed out "I DIFFERENTIATE YOU!"

The new patient calmly looked up and said, "You can differentiate me all you like: I'm e to the x." (CALCULUS)

• The functions are sitting in a bar, chatting (how fast they go to zero at infinity etc.). Suddenly, one cries "Beware! Derivation is coming!"
  All immediately hide themselves under the tables, only the exponential sits calmly on the chair.

The derivation comes in, sees a function and says "Hey, you don't fear me?"
"No, I'am e to x", says the exponential self-confidently.
"Well" replies the derivation "but who says I differentiate along x?" (CALCULUS)

• Big party; every possible function is having fun, chatting and drinking

  this evening.

  In an n-dimensional corner e^x stands bitter and alone.  Near the lonely

  one there's a small group of exponential functions, and 2^x within them

  turns to see e^x on it's corner.

  - Hey, e^x, come-on, integrate yourself - Said 2^x pointing to the group.

  - What for - whispers e^x - it makes no difference

• Math and Alcohol don't mix, so...

  PLEASE DON'T DRINK AND DERIVE

  Then there's every parent's scream when their child walks into the

  room dazed and staggering:

  OH NO...YOU'VE BEEN TAKING DERIVATIVES!!

• The limit as n goes to infinity of sin(x)/n is 6.
  
  Proof: cancel the n in the numerator and denominator.

• In a dark, narrow alley, a function and a differential operator meet:
  "Get out of my way - or I'll differentiate you till you're zero!"
  "Try it - I'm e^x..."
• Same alley, same function, but a different operator:
  "Get out of my way - or I'll differentiate you till you're zero!"
  "Try it - I'm e^x..."
  "Too bad... I'm d/dy."
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