be a rational number with 
and 
.
.
![\[\sum_{k=0}^{n}\binom{2n+1}{2k+1}2^{3k}\]](http://latex.artofproblemsolving.com/d/4/5/d4511807f1e04ac2fa8aae36d65f9e0f5ba11c6f.png)
is not divisible by 
for any integer 
.
is composite, then 
divides 
.
consecutive positive integers is divisible by 
such that 
divides 
.![\[A=\frac{(m+3)^{n}+1}{3m}\]](http://latex.artofproblemsolving.com/f/5/8/f58ed8713c887510943d543c89f83640a546401f.png)
is an integer. Prove that 
is odd.
be divisible by 
.
is divisible by 
and 
be natural numbers having the property ![\[n_{2}\; \vert \; 2^{n_{1}}-1, n_{3}\; \vert \; 2^{n_{2}}-1, \cdots, n_{k}\; \vert \; 2^{n_{k-1}}-1, n_{1}\; \vert \; 2^{n_{k}}-1.\]](http://latex.artofproblemsolving.com/2/0/7/20707ef12d0e08bec8067225a9e9128bbc1d2d45.png)
Show that 
.
.
Y by ahmedosama, UpvoteFarm, Adventure10, Mango247
Show that the polynomial 
can be expressed as the product of two nonconstant polynomials with integer coefficients.
can be expressed as the product of two nonconstant polynomials with integer coefficients.
.![\[\begin{array}{lll}x^{8}+98 x^{4}+1&=&(x^{4}+8x^{2}+1)^{2}-16x^{2}(x^{2}-1)^{2}\\&=& (x^{4}+4x^{3}+8x^{2}-4x+1)(x^{4}-4x^{3}+8x^{2}+4x+1)\end{array}\]](http://latex.artofproblemsolving.com/7/3/d/73d3cf99175184d5dc5fbd22cbe5ec0cd6c549af.png)
does the job.
