# Find the last digit of a series

• MHB
• anemone
In summary, the last digit of $1+2+\cdots+n$ has to be 1 if the last digit of $1^3+2^3+\cdots+n^3$ is 1. This is because $S_3 = S_1^2$ and $S_1\equiv S_3\pmod2$. Additionally, $S_1\equiv3n(n+1)\pmod5$, and by listing the values of $S_1$ for $n\equiv0\dots4\pmod5$, we see that $S_1\equiv1\pmod{10}$. Therefore, the last digit of $1+2+\cdots+n$ must
anemone
Gold Member
MHB
POTW Director
What is the last digit of $1+2+\cdots+n$ if the last digit of $1^3+2^3+\cdots+n^3$ is 1?

[sp]
We have:
\begin{align*}
S_1 &= 1 + \cdots + n = \dfrac{n(n+1)}{2}\\
S_3 &= 1^3 + \cdots + n^3 = \dfrac{n^2(n+1)^2}{4}
\end{align*}
This shows that $S_3 = S_1^2$. Therefore, if $S_3\equiv1\pmod{10}$, then $S_1\equiv\pm1\pmod{10}$.
It is rather obvious that $S_1\equiv S_3\pmod2$.
We may write $S_1\equiv3n(n+1)\pmod5$, since the multiplicative inverse of $2$ is $3$.
We list the value of $S_1$ for $n\equiv0\dots4\pmod5$:
$$\begin{array}{c|c|c|c|c|c} n&0&1&2&3&4\\ \hline S_1\pmod5&0&1&3&1&0 \end{array}$$
and we see that we cannot have $S_1\equiv-1\pmod5$; therefore, $S_1\equiv1\pmod{10}$
[/sp]

We see that the $1^3+2^3+\cdots+n^3 = (1+2+\cdots+n)^2$

we know that the LHS has last digit 1(given) so $(1+2+\cdots+n)$ has last digit 1 or 9.

$(1+2+\cdots+n) = \frac{n(n+1)}{2} = 10k + m$ say for some k and m

so $n(n+1) = 20 k + 2m$

or $4n(n+1) + 1 = 80k + 8m + 1$
or $(2n+1)^2 = 80k + 8m + 1$

if m =9 then 8m + 1 ends with 3. so the square ends with 3. As no square ends with 3 so m cannot be 9. but 1 is possible

So last digit is 1 that is for n of the form 5k +1

Last edited:

## 1. How do I find the last digit of a series?

To find the last digit of a series, you can use a mathematical formula or algorithm. The most common method is to take the last digit of each number in the series and add them together, then take the last digit of the sum. This will give you the last digit of the series.

## 2. Can I use a calculator to find the last digit of a series?

Yes, you can use a calculator to find the last digit of a series. However, it is important to make sure that the calculator has a large enough display to show all the digits in the series. Otherwise, the last digit may be cut off and the answer will be incorrect.

## 3. What if the series has a large number of digits?

If the series has a large number of digits, it may be helpful to use a computer program or software to find the last digit. These programs can handle larger numbers and may have built-in functions for finding the last digit of a series.

## 4. Is there a shortcut for finding the last digit of a series?

Yes, there is a shortcut for finding the last digit of a series. If the series follows a pattern, you can use that pattern to determine the last digit without having to add all the numbers together. For example, if the series is 2, 4, 6, 8, 10, the last digit will always be 0.

## 5. Can I use the last digit of a series to predict future numbers in the series?

No, you cannot use the last digit of a series to predict future numbers. The last digit of a series is based on the previous numbers and does not necessarily follow a pattern or rule that can be used to predict future numbers. Each number in the series should be treated as an individual value and not be used to make predictions.

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