# Congruence in Z(integers) mod n

• capsfan828
In summary, (a + b)5 in Z mod 5 is equivalent to a5 + b5, since 5 is equivalent to 0 in Z mod 5. This can be represented as 5 \equiv 0 mod 5. It is recommended to test this with specific numbers to ensure consistency.
capsfan828

## Homework Statement

In Z mod 5, compute (a + b)5.

## The Attempt at a Solution

Noticing that (a+b)5 = a5+5(a4*b+2*a3*b2+2*a2*b3+a*b4)+b5. Since 5=0 in Z mod 5, it follows that 0(a4*b+2*a3*b2+2*a2*b3+a*b4)=0 and hence (a+b)5=a5+b5.

I am just wondering if it is correct for me to say 5=0 in Z mod 5 and just substitute 0 in for 5?

$$\equiv$$Sounds reasonable to me. Why don't you test this with a couple of numbers to see if your results are consistent with what you've found?

BTW, instead of saying 5=0 in Z mod 5, you can say 5 $\equiv$ 0 mod 5. That 3-bar equals sign means "is equivalent to".

thanks for the response, much appreciated

## 1. What is "congruence" in Z(integers) mod n?

Congruence in Z(integers) mod n is a mathematical concept that describes the relationship between two integers when they have the same remainder when divided by a given integer n. In other words, two integers are congruent if they have the same value in the same "clock" of integers, starting at 0 and ending at n-1.

## 2. How is congruence in Z(integers) mod n different from regular division?

Regular division focuses on finding the quotient and remainder when dividing two integers. Congruence in Z(integers) mod n, on the other hand, focuses on the relationship between two integers when divided by a given integer n. It disregards the quotient and only considers the remainder.

## 3. What is the notation used to express congruence in Z(integers) mod n?

Congruence in Z(integers) mod n is typically expressed using the notation "a ≡ b (mod n)", where a and b are the two integers being compared, and n is the given modulus. The triple bar symbol ≡ represents congruence, and the (mod n) indicates the modulus being used.

## 4. How is congruence in Z(integers) mod n useful in solving mathematical problems?

Congruence in Z(integers) mod n is useful in solving mathematical problems that involve finding patterns and relationships between integers. It is also used in cryptography, where it helps to encode and decode messages by using modular arithmetic.

## 5. Can two integers be congruent in one modulus but not congruent in another modulus?

Yes, two integers can be congruent in one modulus but not congruent in another modulus. This is because the modulus determines the "clock" of integers being used, and different moduli will have different remainders for the same two integers. For example, 7 ≡ 14 (mod 7) but 7 ≡ 0 (mod 2).

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