Question

Докажите, что при любом натуральном n значение выражения (6n-1) - (2n-2) при делении на 4 даёт остаток, равный 1.

Answer 1

(6n-1)-(2n-2)=6n-1-2n+2=4n+1
если домножить любое число на 4,то оно будет делится
если к 4n добавить 1,то остаток=1

Answer 2

Proof that the expression (6n-1) - (2n-2) gives a remainder of 1 when divided by 4

To prove that the expression (6n-1) - (2n-2) gives a remainder of 1 when divided by 4 for any natural number n, we can use modular arithmetic.

Let's simplify the expression step by step:

(6n-1) - (2n-2)

Simplifying further:

6n - 1 - 2n + 2

Combining like terms:

4n + 1

Now, let's consider the expression (4n + 1) modulo 4. The modulo operation gives us the remainder when the expression is divided by 4.

To prove that the remainder is 1, we need to show that (4n + 1) can be written as 4k + 1, where k is an integer.

Let's rewrite (4n + 1) as (4n + 0 + 1):

4n + 0 + 1

Now, we can group the terms:

(4n + 0) + 1

Since 4n is divisible by 4, we can rewrite it as 4k, where k is an integer:

4k + 1

Therefore, we have shown that (4n + 1) can be written as 4k + 1, where k is an integer. This means that the expression (6n-1) - (2n-2) gives a remainder of 1 when divided by 4 for any natural number n.

Note: The proof above is based on the properties of modular arithmetic and the fact that the expression (6n-1) - (2n-2) can be simplified to 4n + 1.