Feel stuck and not sure how to start...?

Question

Let n>1 be an integer and suppose a,b є Zn+1 = {qn+1 : q є Z}.
Show that the product ab also belongs to Zn+1.
(Z is the set of all integers)
I'm not really sure how to go about starting this...

Answer 1

If a and b are in "Zn+1", then given an integer n>1, a=sn+1 and b=tn+1, for some integers s and t. This means ab = (sn+1)(tn+1) = stn² + sn + tn + 1, or (stn+s+t)n + 1. Let q = (stn+s+t). If you can show that q is always an integer, then it shows ab can be written in the form of qn+1, and thus ab is in the set Zn+1.

Answer 2

by induction:
name the set A:={qn+1:q є z}
1 = 0n+1, so 1 is in A.
Let x,y be a number in the set A. If n is in A then x+1 is in A, by definition of the set. Then assume x + y is in A. So x +(y+1) = (x+y)+1, so by induction, A is closed under addition.

Let B be the set B:={mєN: n*m є N}
n*1 = n, so 1 є B. If some m > 1 є B, then n(m+1) = nm+n by closure of addition. So m+1єB and then m*n is in B by induction.

Answer 3

If a and b are elements of that set, then there are j and k such that
a = kn+1 for a given n>1 and
b = jn+1 for given n>1.

a*b = (kn+1)(jn+1) = kjn^2 + kn + jn + 1 = (kjn + k + j)n + 1
And the thing in parentheses is an integer. So you're done.