# Prove that ## a^{4}\equiv 0 ## or ## 1\pmod {5} ##?

• Math100
In summary, the conversation discusses the proof of the assertion that for any integer a, a^4 is congruent to 0 or 1 mod 5. The proof involves showing that a^4 is congruent to 0, 1, 1, 1, or 1 mod 5, and thus concluding that a^4 is always congruent to 0 or 1 mod 5 for any integer a. It also leads to the question of whether a similar proof can be applied for a prime number p and any integer n where p does not divide n.
Math100
Homework Statement
Prove the assertion below:
For any integer ## a ##, ## a^{4}\equiv 0 ## or ## 1\pmod {5} ##.
Relevant Equations
None.
Proof:

Let ## a ## be any integer.
Then ## a\equiv 0, 1, 2, 3 ## or ## 4\pmod {5} ##.
Note that ## a\equiv b\pmod {n}\implies a^{4}\equiv b^{4}\pmod {n} ##.
This means ## a^{4}\equiv 0, 1, 16, 81 ## or ## 256\pmod {5}\implies a^{4}\equiv 0, 1, 1, 1 ## or ## 1\pmod {5} ##.
Thus ## a^{4}\equiv 0 ## or ## 1\pmod {5} ##.
Therefore, ## a^{4}\equiv 0 ## or ## 1\pmod {5} ## for any integer ## a ##.

Delta2 and fresh_42
Math100 said:
Homework Statement:: Prove the assertion below:
For any integer ## a ##, ## a^{4}\equiv 0 ## or ## 1\pmod {5} ##.
Relevant Equations:: None.

Proof:

Let ## a ## be any integer.
Then ## a\equiv 0, 1, 2, 3 ## or ## 4\pmod {5} ##.
Note that ## a\equiv b\pmod {n}\implies a^{4}\equiv b^{4}\pmod {n} ##.
This means ## a^{4}\equiv 0, 1, 16, 81 ## or ## 256\pmod {5}\implies a^{4}\equiv 0, 1, 1, 1 ## or ## 1\pmod {5} ##.
Thus ## a^{4}\equiv 0 ## or ## 1\pmod {5} ##.
Therefore, ## a^{4}\equiv 0 ## or ## 1\pmod {5} ## for any integer ## a ##.
You have shown even more: ##a^4\equiv 1 \pmod 5## whenever ##5\nmid a.##

Math100
fresh_42 said:
You have shown even more: ##a^4\equiv 1 \pmod 5## whenever ##5\nmid a.##
… which directly brings us to the following question that the OP might want to attempt to answer:

If p is prime and ##p \nmid n##, does ##n^{p-1} \equiv 1 \pmod p##?

Last edited:
SammyS

## 1. What does the notation "a^{4}\equiv 0" mean?

The notation "a^{4}\equiv 0" means that the number a raised to the fourth power is congruent to 0 modulo 5. In other words, when a is divided by 5, the remainder is 0.

## 2. How do you prove that a^{4}\equiv 0 or 1\pmod {5}?

To prove that a^{4}\equiv 0 or 1\pmod {5}, we can use the Fermat's Little Theorem which states that if p is a prime number, then a^{p-1}\equiv 1\pmod {p}. In this case, p=5, so a^{4}\equiv 1\pmod {5} if a is not divisible by 5. If a is divisible by 5, then a^{4}\equiv 0\pmod {5}.

## 3. Can you give an example to illustrate this statement?

Sure, let's take a=3. Then 3^{4}\equiv 81\equiv 1\pmod {5} since 81 is divisible by 5. Now, let's take a=6. Then 6^{4}\equiv 1296\equiv 1\pmod {5} since 1296 is also divisible by 5. These examples show that a^{4}\equiv 1\pmod {5} for any number that is not divisible by 5.

## 4. How does this relate to the concept of congruence modulo?

The statement a^{4}\equiv 0 or 1\pmod {5} is an example of congruence modulo, which is a mathematical concept that compares two numbers by dividing them by a common divisor and looking at the remainder. In this case, we are comparing a^{4} and 0 or 1 modulo 5, meaning we are looking at the remainder when these numbers are divided by 5.

## 5. Why is it important to prove this statement?

Proving that a^{4}\equiv 0 or 1\pmod {5} is important because it helps us understand the properties of numbers and their relationships with each other. It also has practical applications in fields such as cryptography and number theory. Additionally, this statement is a special case of a more general theorem (Fermat's Little Theorem) which has many important applications in mathematics and science.

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