Question

Докажите что выражение n^5+4n делиться на 5,где (n-натуральное число)

Answer 1

N^5+4n=n(n^4+4)  Если n=5k, то утверждение очевидно. В остальных случаях n=5k+1 или n=5k-1, и 5k+2 или 5k-2 выражения  (5k+1)^4 дает остаток один при делении на 5, (5k-1)^4 дает остаток один при делении на 5, (5k-2)^4 дает остаток один при делении на 5, (5k+1)^4 дает остаток один при делении на 5. Таким образом после прибавления к числу, дающему остаток один при делении на 5 числа 4 во всех случаях получаем число, которое делится на 5.  

Answer 2

Proof that the Expression n^5 + 4n is Divisible by 5

To prove that the expression n^5 + 4n is divisible by 5 for n being a natural number, we can use mathematical induction.

Base Case (n=1): When n=1, the expression becomes 1^5 + 4*1 = 1 + 4 = 5, which is divisible by 5.

Inductive Step: Assuming the expression n^5 + 4n is divisible by 5 for some natural number k, we need to show that it holds for k+1 as well.

Let's consider the expression for k+1: (k+1)^5 + 4(k+1) = k^5 + 5k^4 + 10k^3 + 10k^2 + 5k + 1 + 4k + 4 = (k^5 + 4k) + 5(k^4 + 2k^3 + 2k^2 + k) + 5

By the inductive assumption, k^5 + 4k is divisible by 5. Additionally, 5(k^4 + 2k^3 + 2k^2 + k) is also divisible by 5. Therefore, the entire expression is divisible by 5.

Hence, by mathematical induction, we have proved that the expression n^5 + 4n is divisible by 5 for n being a natural number.