003/POTW 297 Secondary school/ High school students

https://mathhelpboards.com/potw-secondary-school-high-school-students-35/problem-week-297-jan-17th-2018-a-23325.html#post104312

Problem:
if $k_1,k_2,\cdots k_{15}$ are the roots of the equation $x^{15} - 2x^{14} + 3 x^{13} + \cdots + 15x-16= 0$ evaluate $(1+k_1)(1+k_2)\cdots(1+k_{15})$

Because $k_1,k_2,\cdots k_{15}$ are the root

Hence $(x-k_1)(x-k_2)\cdots(x-k_{15}) = x^{15} - 2x^{14} + 3 x^{13} + \cdots + 15x-16$ as RHS is a degree 15 polynomial and coefficient is 1

putting $x= -1 $ we get
$(-1-k_1)(-1-k_2)\cdots(-1-k_{15}) = -1 - 2 -3 - \cdots - 15-16= -136$
or
$(-1)^{15}(1+k_1)(1+k_2),\cdots(-1+k_{15}) = -136$
or $(1+k_1)(1+k_2)\cdots(-1+k_{15}) = 136$

002/ POTW 297 University

http://mathhelpboards.com/potw-university-students-34/problem-week-297-jan-18-2018-a-23332.html#post104342l


Problem 
For which nonnegative integers n and k $(k+1)^n +(k +2)^n +(k +3)^n + (k+4)^n +(k +5)^n$ is divisible by 5.
Solution
Let us define
$f(k,n) = (k+1)^n +(k +2)^n +(k +3)^n + (k+4)^n +(k +5)^n$

Hence $f(k+1,n) - f(k,n  ) = (k+6)^n - (k+1)^n$ and it is divisible by $(k+6) - (k+1)$ or 5. this is independent of n

So $f(k+1,n)$ is divisible by 5 iff $f(k,n)$ is divisible by 5

Further as 5 is prime we have as per Fermat's Little theorem $x^5=x$ for all x .

Hence if $f(k,p)$ is divisible by 5 then $f(k,p+5)$ is divisible by 5

So we need to check for $f(0,0),f(0,1),f(0,2),f(0,3),f(0,4)$ and we get
$f(0,0)=5,f(0,1) = 15, f(0,2) = 55, f(0,3) = 225, f(0,4) = 979$

All except $f(0,4)$ is divisible by 5

So it is divisible by 5 for all k and  n which is not of the form 5p+ 4 for non negative k and p  

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