need a easy way to solve it

For which integers are x^5+4x^4+3x+1 divisible by 5?

i have solved this prob and there is none but i did it on a "hard way"

i did make it to x(x^3(x+4)+3)+1 and try put like x to 1,2,3 etc.. is there any easy way to solve it?

(sorry for bad english:/)

Re: need a easy way to solve it

Re: need a easy way to solve it

Let's plase all of the integers into one of 5 different classes:

x = 5a + 0, or

x = 5a + 1, or

x = 5a + 2, or

x = 5a + 3, or

x = 5a + 4

for each type of number I would try substituting for x in your original equation and determining if it is possible for the result to be divisible by 5. For example:

Let X = 5a + 0. Yor equation becomes:

$\displaystyle (5a)^5+4(5a)^4+3(5a)+1 = 5(5^4a^5+4.5^3.a^4+3a)+1$ and this cannot be divisible by 5 so the enture class of numbers x=5a+0 cannot give a result that is divisible by 5.

Let X = 5a + 1. Yor equation becomes:

$\displaystyle (5a+1)^5+4(5a+1)^4+3(5a+1)+1 $

When you multiply out the first bracket only the last term is not divisible by 5, the last term is 1

When you multiply out the second bracket only the last term is not divisible by 5, the last term is 4

When you multiply out the third bracket only the last term is not divisible by 5, the last term is 3

so we can say that for some k

$\displaystyle (5a+1)^5+4(5a+1)^4+3(5a+1)+1 = 5k + 1+4+3+1=5k'+4$ and this cannot be divisible by 5 so the enture class of numbers x=5a+1 cannot give a result that is divisible by 5.

Now you can examine the remaining three classes of integers in the same way.

Re: need a easy way to solve it

$\displaystyle f({x})\text{=}x^5+4x^4+3x+1$

for k=1,2,3,....n

$\displaystyle \text{Mod}(f(k), 5) = \{4,3,2,1,1, 4,3,2,1,1, 4,3,2,1,1, ....\}$

Re: need a easy way to solve it

Just check x = 0,1,2,3,4. If x = k works, then any x that is congruent to k (mod 5) also works.