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Dealing with contradiction: Show that for any positive integer a and any prime p, if p is divides a, then p does not divide a+1.

Show that the polynomial p(x)= x^4 + 2x^2 + 2x + 2 cannot be expressed as the product of two quadratic polynomials of form x^2 + ax + b and x^2 + cx +d where a,b,c, and d are any integers.

Show that log5(8) is irrational.