# Derive the following congruence....

• Math100
In summary, the conversation discusses the application of Fermat's theorem and the resulting equations for a number that is equal to 3 times 5. It is shown that for all values of a, the equation a^21 ≡ a (mod 15) holds true. The participant suggests an alternative way of writing the equation for clarity, and notes that their confusion may be due to the time of day.
Math100
Homework Statement
Derive the following congruence:
## a^{21}\equiv a\pmod {15} ## for all ## a ##.
[Hint: By Fermat's theorem, ## a^{5}\equiv a\pmod {5} ##.]
Relevant Equations
None.
Proof:

Observe that ## 15=3\cdot 5 ##.
Applying the Fermat's theorem produces:
## a^{3}\equiv a\pmod {3} ## and ## a^{5}\equiv a\pmod {5} ##.
Thus
\begin{align*}
&(a^{3})^{7}\equiv a^{7}\pmod {3}\implies a^{21}\equiv [(a^{3})^{2}\cdot a]\pmod {3}\implies a^{21}\equiv a\pmod {3}\\
&(a^{5})^{4}\equiv a^{4}\pmod {5}\implies a^{20}\equiv a^{4}\pmod {5}\implies a^{21}\equiv a\pmod {5}.\\
\end{align*}
Therefore, ## a^{21}\equiv a\pmod {15} ## for all ## a ##.

Maybe you should write ##a^{21}\equiv a^7\equiv a^3\cdot a^3 \cdot a\equiv a\cdot a\cdot a\equiv a^3\equiv a\pmod{3}.## Or at least ##(a^3)^2\cdot a\equiv a^2\cdot a\equiv a^3\equiv a\pmod{3}.##

I stumbled upon ##a^6\cdot a\equiv a\pmod{3}## which was not immediately clear (to me).

But again: might be due to local time.

Math100
fresh_42 said:
Maybe you should write ##a^{21}\equiv a^7\equiv a^3\cdot a^3 \cdot a\equiv a\cdot a\cdot a\equiv a^3\equiv a\pmod{3}.## Or at least ##(a^3)^2\cdot a\equiv a^2\cdot a\equiv a^3\equiv a\pmod{3}.##

I stumbled upon ##a^6\cdot a\equiv a\pmod{3}## which was not immediately clear (to me).

But again: might be due to local time.
Sorry, it's my fault.

Math100 said:
Sorry, it's my fault.
Not at all. I'm just a little bit slow at night time. That's not your fault. It only proves that the Earth is not flat.

Math100

## What is a congruence in mathematics?

A congruence in mathematics refers to the relationship between two objects that have the same size and shape. In other words, two objects are congruent if they are identical in every way, including their size, shape, and orientation.

## What does it mean to derive a congruence?

Deriving a congruence means to prove or demonstrate that two objects are congruent using a set of mathematical principles and rules. This involves showing that the two objects have the same size, shape, and orientation through a series of logical steps.

## What are the steps involved in deriving a congruence?

The steps involved in deriving a congruence may vary depending on the specific problem, but generally involve identifying and using properties of congruent objects, such as side lengths, angles, and symmetry. This may also involve using geometric theorems and postulates to make logical deductions.

## What is the purpose of deriving a congruence?

The purpose of deriving a congruence is to prove that two objects are identical in size, shape, and orientation. This can be useful in solving mathematical problems and providing a deeper understanding of geometric concepts.

## What are some common methods for deriving a congruence?

Some common methods for deriving a congruence include using the properties of congruent triangles, such as SSS (side-side-side), SAS (side-angle-side), and ASA (angle-side-angle). Other methods may involve using properties of parallel lines, perpendicular lines, and symmetry.

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