For ## n\geq 1 ##, use congruence theory to establish....

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In summary: The title can just be "Prove that ##7\mid (5^{2n} + 3\cdot 2^{5n-2})##" or "Prove divisibility statement with congruence theory" or something similar. This will help others who are browsing the forum and looking for specific topics to find your threads and potentially provide answers. Thank you.
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Math100
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Homework Statement
For ## n\geq 1 ##, use congruence theory to establish the following divisibility statement:
## 7\mid (5^{2n}+3\cdot 2^{5n-2}) ##.
Relevant Equations
None.
Proof:

Let ## n\geq 1 ## be a natural number.
Note that ## 5^{2}\equiv 4\pmod 7\implies (5^{2})^{n}\equiv 4^{n}\pmod {7} ##.
Now observe that \begin{align*} (3\cdot 2^{5n-2})&\equiv (3\cdot 2^{3}\cdot 2^{5n-5})\pmod {7}\\
&\equiv [3\cdot 2^{3}\cdot (2^{5})^{n-1}]\pmod {7}\\
&\equiv [24\cdot (2^{5})^{n-1}]\pmod {7}\\
&\equiv (3\cdot 4^{n-1})\pmod {7}\\
&\equiv (-4\cdot 4^{n-1})\pmod {7}\\
&\equiv -(4^{n})\pmod {7}.
\end{align*}
Thus ## 5^{2n}+3\cdot 2^{5n-2}\equiv (4^{n}-4^{n})\pmod 7\equiv 0\pmod 7 ##.
Therefore, ## 7\mid (5^{2n}+3\cdot 2^{5n-2}) ## for ## n\geq 1 ##.
 
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Math100 said:
Homework Statement:: For ## n\geq 1 ##, use congruence theory to establish the following divisibility statement:
## 7\mid (5^{2n}+3\cdot 2^{5n-2}) ##.
Relevant Equations:: None.

Proof:

Let ## n\geq 1 ## be a natural number.
Note that ## 5^{2}\equiv 4\pmod 7\implies (5^{2})^{n}\equiv 4^{n}\pmod {7} ##.
Now observe that \begin{align*} (3\cdot 2^{5n-2})&\equiv (3\cdot 2^{3}\cdot 2^{5n-5})\pmod {7}\\
&\equiv [3\cdot 2^{3}\cdot (2^{5})^{n-1}]\pmod {7}\\
&\equiv [24\cdot (2^{5})^{n-1}]\pmod {7}\\
&\equiv (3\cdot 4^{n-1})\pmod {7}\\
&\equiv (-4\cdot 4^{n-1})\pmod {7}\\
&\equiv -(4^{n})\pmod {7}.
\end{align*}
Thus ## 5^{2n}+3\cdot 2^{5n-2}\equiv (4^{n}-4^{n})\pmod 7\equiv 0\pmod 7 ##.
Therefore, ## 7\mid (5^{2n}+3\cdot 2^{5n-2}) ## for ## n\geq 1 ##.

Correct, up to typos in the last line of the align environment which I corrected. You had { ... ) which does not match and ##-4\cdot 4^{n-1}=(-4^{n-1})## which was wrong.
 
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Thank you!
 
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@Math 100, you have started several threads with very generic and uninformative titles such as "For ## n\geq 1 ##, use congruence theory to establish?". For future threads, please provide titles that indicate what it is you're trying to prove. For this thread, a better title would be "Prove that ## 7\mid (5^{2n}+3\cdot 2^{5n-2}) ##".
It's not necessary to give all of the details in the thread, such as ##n \ge 1##.
 

FAQ: For ## n\geq 1 ##, use congruence theory to establish....

1. What is congruence theory?

Congruence theory is a branch of mathematics that deals with the properties and relationships of congruent objects. Two objects are said to be congruent if they have the same size and shape. In congruence theory, we use modular arithmetic to study the properties of congruent objects.

2. How is congruence theory used to establish a statement?

In congruence theory, we use modular arithmetic to establish a statement by showing that two objects are congruent modulo a certain number. This means that the remainder when dividing the two objects by the specified number is the same, indicating that they have the same properties and relationships.

3. What is the significance of the statement "for n ≥ 1" in congruence theory?

The statement "for n ≥ 1" in congruence theory means that we are considering all positive integers as potential values for n. This allows us to establish a general statement that holds true for any value of n greater than or equal to 1.

4. Can you provide an example of using congruence theory to establish a statement?

Sure, for example, we can use congruence theory to establish that for any integer n, n^2 is congruent to 1 modulo 3. This means that the remainder when dividing n^2 by 3 will always be 1. We can prove this by considering the three possible cases for n: n ≡ 0 (mod 3), n ≡ 1 (mod 3), or n ≡ 2 (mod 3) and showing that in each case, n^2 ≡ 1 (mod 3).

5. What are some real-world applications of congruence theory?

Congruence theory has many practical applications, including in cryptography, computer science, and engineering. It is used in the design and analysis of algorithms, as well as in the development of secure communication protocols. It is also used in the study of patterns and symmetries in art and design, as well as in the construction of geometric shapes and structures.

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