Prove that ## a^{3}\equiv 0, 1 ##, or ## 6\pmod {7} ##?

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In summary, using the definition of congruence and the fact that taking the cube of any integer will result in a congruent value, it can be proven that for any integer ## a ##, ## a^{3}\equiv 0, 1, ## or ## 6\pmod {7} ##.
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Math100
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Homework Statement
Prove the assertions below:
For any integer ## a ##, ## a^{3}\equiv 0, 1, ## or ## 6\pmod {7} ##.
Relevant Equations
None.
Proof:

Let ## a ## be any integer.
Then ## a\equiv 0, 1, 2, 3, 4, 5, ## or ## 6\pmod {7} ##.
Note that ## a\equiv b\pmod {n}\implies a^{3}\equiv b^3\pmod{n} ##.
This means ## a^{3}\equiv 0, 1, 8, 27, 64, 125 ## or ## 216\pmod{7}\implies a^{3}\equiv 0, 1, 1, 6, 1, 6 ## or ## 6\pmod {7} ##.
Thus ## a^{3}\equiv 0, 1 ## or ## 6\pmod {7} ##.
Therefore, ## a^{3}\equiv 0, 1, ## or ## 6\pmod {7} ## for any integer ## a ##.
 
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  • #2
Math100 said:
Homework Statement:: Prove the assertions below:
For any integer ## a ##, ## a^{3}\equiv 0, 1, ## or ## 6\pmod {7} ##.
Relevant Equations:: None.

Proof:

Let ## a ## be any integer.
Then ## a\equiv 0, 1, 2, 3, 4, 5, ## or ## 6\pmod {7} ##.
Note that ## a\equiv b\pmod {n}\implies a^{3}\equiv b^3\pmod{n} ##.
This means ## a^{3}\equiv 0, 1, 8, 27, 64, 125 ## or ## 216\pmod{7}\implies a^{3}\equiv 0, 1, 1, 6, 1, 6 ## or ## 6\pmod {7} ##.
Thus ## a^{3}\equiv 0, 1 ## or ## 6\pmod {7} ##.
Therefore, ## a^{3}\equiv 0, 1, ## or ## 6\pmod {7} ## for any integer ## a ##.
Correct.
 
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FAQ: Prove that ## a^{3}\equiv 0, 1 ##, or ## 6\pmod {7} ##?

What does it mean for a number to be equivalent to 0, 1, or 6 mod 7?

When we say that a number is equivalent to 0, 1, or 6 mod 7, it means that when the number is divided by 7, the remainder will be either 0, 1, or 6. This is also known as the number being congruent to 0, 1, or 6 mod 7.

Why is it important to prove that a number is equivalent to 0, 1, or 6 mod 7?

Proving that a number is equivalent to 0, 1, or 6 mod 7 can be helpful in various mathematical proofs and calculations. It can also help in simplifying complex equations and finding patterns in numbers.

How can we prove that a number is equivalent to 0, 1, or 6 mod 7?

To prove that a number is equivalent to 0, 1, or 6 mod 7, we can use the division algorithm to divide the number by 7 and check the remainder. If the remainder is 0, 1, or 6, then the number is equivalent to 0, 1, or 6 mod 7, respectively.

Can a number be equivalent to more than one value mod 7?

No, a number can only be equivalent to one value mod 7. This is because when we divide a number by 7, the remainder can only be one of the values 0, 1, 2, 3, 4, 5, or 6. Therefore, a number can only be congruent to one of these values mod 7.

What are some real-world applications of understanding modular arithmetic and proving congruence?

Understanding modular arithmetic and proving congruence can be useful in fields such as cryptography, computer science, and engineering. It is also used in solving problems related to calendars, time zones, and repeating patterns in nature.

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