Is there any last digit in mathematics? if yes than what that digit is?
Answers
Answer:
If there is a sign specifically devoted to that, I do not know it. It would probably be a pretty localized usage of it, though. However, that is not to say that there are no ways to express the notion other than prose.
In fact, there are at least three ways to say that with symbols.
However, first, we have to define what “the last digit” of a number means. The decimal system that I assume you are referring to is not anything special in number theory, which is why no-one is really interested in specifically the decimal last digit. You will note, however, that the last digit of a number written in decimal is exactly the remainder of dividing the number by 10. As per the Division Theorem, the way to state that is:
43=4∗10+343=4∗10+3
Another way of putting the above into symbols is:
43≡3(mod10)43≡3(mod10)
The above reads “43 is congruent to 3 modulo 10”. Mathematicians on the whole don’t really care how you write this, as long as it’s clear from context that the you are trying to say that 3 is the last digit of 43.
I have seen some mathematicians shorten the above into a subscript, especially when they are dealing with many congruences in different radixes:
43≡10343≡103
But don’t try using that unless you say what you mean by it in a paper.
Answer:
the first 4 factorials:
1! = 1 last digit of 1
2! = 2 last digit of 2
3! = 6 last digit of 6
4! = 24 last digit of 4
Summing those last digits gives us 13, which has a last digit of 3 which will be the last digit of the number being raised to the 33rd power.
3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81
It should now be clear that the last digits of the powers of 3 will cycle through the digits {3,9,7,1}. This means that the 32nd power of 3 will have a last digit of 1 and the 33rd will have a last digit of 3.
So the number you specified will end in a digit of 3. Whew!