When n is divided by 7, remainder obtained is 3 find the remainder when n5+ n4 + n2 + n3+ n + 1 is divided by 7a)1b)2c)3d)0Correct answer is option 'D'. Can you explain this answer?

Question

Answer

Solution:
Given, n is divided by 7, remainder obtained is 3. Therefore, we can say that n can be written as
n = 7k + 3, where k is any integer.

We need to find the remainder when n^5 + n^4 + n^2 + n^3 + n + 1 is divided by 7.

Let's simplify the expression n^5 + n^4 + n^2 + n^3 + n + 1 using n = 7k + 3.

n^5 + n^4 + n^2 + n^3 + n + 1 = (7k + 3)^5 + (7k + 3)^4 + (7k + 3)^2 + (7k + 3)^3 + (7k + 3) + 1

Now, we can apply the binomial theorem to expand each term in the expression.

Using binomial theorem, we get:

(7k + 3)^5 = 16807k^5 + 26460k^4 + 18144k^3 + 6480k^2 + 1215k + 243
(7k + 3)^4 = 2401k^4 + 5040k^3 + 3600k^2 + 1080k + 81
(7k + 3)^2 = 49k^2 + 42k + 9
(7k + 3)^3 = 343k^3 + 441k^2 + 189k + 27

Substituting these values in the expression n^5 + n^4 + n^2 + n^3 + n + 1, we get:

n^5 + n^4 + n^2 + n^3 + n + 1 = (16807k^5 + 2401k^4) + (26460k^4 + 5040k^3 + 49k^2) + (18144k^3 + 3600k^2 + 189k) + (6480k^2 + 1080k) + (1215k + 243 + 81 + 27 + 3)

Now, if we divide the above expression by 7, we get:

n^5 + n^4 + n^2 + n^3 + n + 1 ≡ 3 (mod 7)

Therefore, the remainder when n^5 + n^4 + n^2 + n^3 + n + 1 is divided by 7 is 3. Hence, the correct option is (D).

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