Prove a^5 congruent to a (mod 15)

Thread starter sty2004
Start date Feb 3, 2010

In summary, "mod" stands for "modulo" and is the remainder when a number is divided by another number. Proving congruence between two numbers, such as a^5 and a (mod 15), is important for making predictions and solving equations with large numbers. To prove this congruence, we need to show that a^5 and a have the same remainder when divided by 15. If a is not a multiple of 15, we can still prove the congruence using the properties of modular arithmetic. This congruence can also hold for other numbers besides 15, with the specific methods varying based on the value of n.

Feb 3, 2010

sty2004

Homework Statement

Prove that a⁵ [tex]\equiv[/tex]a (mod 15) for every integer a.

Homework Equations

The Attempt at a Solution

I do not know how to show a⁵-a is divisible by 15

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Feb 4, 2010

Mandark

[tex]a^5 - a = (a-1)a(a+1)(a^2 + 1)[/tex] can you show that this expression is divisible by 3 and by 5 individually? This would imply it's divisible by 15.

What does "mod" mean in this context?

"Mod" is short for "modulo" and refers to the remainder when a number is divided by another number. In this case, it is the remainder when a^5 is divided by 15.

Why is it important to prove this congruence?

Proving congruence between two numbers is important because it allows us to make predictions and solve equations with large numbers more easily. It also has applications in fields such as cryptography and computer science.

How do you prove a^5 congruent to a (mod 15)?

To prove this congruence, we need to show that a^5 and a have the same remainder when divided by 15. This can be done by using the properties of modular arithmetic and manipulating the exponents and bases of the numbers.

What if a is not a multiple of 15?

If a is not a multiple of 15, we can still prove the congruence by using the fact that a^5 and a will have the same remainder when divided by 15. This is because the remainder of a^5 will depend only on the remainder of a when divided by 15.

Can this congruence hold for other numbers besides 15?

Yes, this congruence can hold for other numbers besides 15. In general, we can prove a^5 congruent to a (mod n) for any positive integer n. However, the specific steps and methods used may vary depending on the value of n.