Verify that ## a^{5} ## and ## a ## have the same units digit

• Math100
In summary, the proof demonstrates that for any integer ##a##, the powers ##a^5## and ##a## have the same units digit. This is shown through the application of Fermat's theorem and the use of modular arithmetic.
Math100
Homework Statement
For any integer ## a ##, verify that ## a^{5} ## and ## a ## have the same units digit.
Relevant Equations
None.
Proof:

Let ## a ## be any integer.
Applying the Fermat's theorem produces:
## a^{2}\equiv a\pmod {2}, a^{5}\equiv a\pmod {5} ##.
Observe that
\begin{align*}
&a^{4}\equiv a^{2}\pmod {2}\equiv a\pmod {2}\\
&a^{5}\equiv a^{2}\pmod {2}\equiv a\pmod {2}.\\
\end{align*}
This means ## a^{5}\equiv a\pmod {10} ##.
Suppose ## 0\leq k<10 ##.
Then ## a^{5}-k\equiv (a-k)\pmod {10} ##.
Thus ## a^{5}-k\equiv 0\pmod {10}\implies a-k\equiv 0\pmod {10} ##.
Therefore, ## a^{5} ## and ## a ## have the same units digit for any integer ## a ##.

WWGD and fresh_42
Math100 said:
Homework Statement:: For any integer ## a ##, verify that ## a^{5} ## and ## a ## have the same units digit.
Relevant Equations:: None.

Proof:

Let ## a ## be any integer.
Applying the Fermat's theorem produces:
## a^{2}\equiv a\pmod {2}, a^{5}\equiv a\pmod {5} ##.
Observe that
\begin{align*}
&a^{4}\equiv a^{2}\pmod {2}\equiv a\pmod {2}\\
&a^{5}\equiv a^{2}\pmod {2}\equiv a\pmod {2}.\\
\end{align*}
This means ## a^{5}\equiv a\pmod {10} ##.
Suppose ## 0\leq k<10 ##.
Then ## a^{5}-k\equiv (a-k)\pmod {10} ##.
Thus ## a^{5}-k\equiv 0\pmod {10}\implies a-k\equiv 0\pmod {10} ##.
Therefore, ## a^{5} ## and ## a ## have the same units digit for any integer ## a ##.
Fine. Only one comment. As long as you consider the same modulus, it isn't necessary to repeat it. I.e. ##a^5\equiv a^2\equiv a \pmod{5}## is sufficient. All are modulo ##5.## It is different if we write ##a^5\equiv a^2\pmod{5} \Longrightarrow a^5\equiv a\pmod{5}.## The implication interrupts the calculation, the ##\equiv ## sign has the same meaning as ##=.##

Math100

1. What does it mean for two numbers to have the same units digit?

Having the same units digit means that the numbers have the same number in the ones place, or the rightmost digit. For example, both 23 and 43 have a units digit of 3.

2. Why is it important to verify that two numbers have the same units digit?

Verifying that two numbers have the same units digit can help check for errors in calculations or conversions. It can also provide insight into patterns and relationships between numbers.

3. How can I verify that two numbers have the same units digit?

To verify that two numbers have the same units digit, you can simply look at the rightmost digit of each number. If they are the same, then the numbers have the same units digit. Additionally, you can use modular arithmetic or perform long division to check for the same remainder.

4. Can two numbers with different units have the same units digit?

Yes, two numbers with different units can have the same units digit. The units digit only refers to the rightmost digit of a number, regardless of its units. For example, both 5 meters and 5 liters have a units digit of 5.

5. What are some real-world applications of verifying units digits?

Verifying units digits can be useful in various fields such as engineering, physics, and finance. For example, engineers may use it to check for errors in calculations for measurements, while physicists may use it to analyze patterns in data. In finance, it can be used to check for errors in currency conversions or to identify repeating patterns in stock prices.

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