(a) Let x, y, z be integers such that x+ y +z = 0. Show that if d is an integer
ID: 3115033 • Letter: #
Question
(a) Let x, y, z be integers such that x+ y +z = 0. Show that if d is an integer which divides two of r, y, z, then it also divides the third of these numbers. Note: your proof should (b) State a generalization of part (a) to n integers ai, a2,..., an whose sum is zero. No proof (c) Suppose that the quadratic equation r2 + br + c = 0 has an integer solution, r. Prove involve three cases is required. to assume in the proof that 0. polynomial A monic polynomial is one in which the coefficient of the highest power of (d) State a generalization of part (c) to the equation p(z)-0, where p(x) is the monic p(x)az -1az2+... + an-1z a equals 1Explanation / Answer
x+y+z=0 has been given;
Also, it is assumed that 'x' and 'y' are divisible by an interger 'd'
We are to prove that 'z' will also be divisble by 'd'
Since 'x' is divisible by 'd' we can write
x= k1 * d
similarly
y= k2*d
(here k1 & k2 are two integers that are quotients when x and y are divided by 'd' respectively)
substituting this in the equation x+y+z=0 we will get :
k1d + k2d + z=0
z= -k1d - k2d
z= d(-k1-k2)
z/d = (-k1-k2) since k1 and k2 are integers, we can say that -k1-k2 will also be an integer
Let's say -k1-k2 = k3
z= d*k3
z/d=k3
Hence we see that 'z' too is divisible by 'd'
Hence, we have assumed 'x' and 'y' to be divisible by 'd' and proven that 'z' too will be divisible by 'd'
Similalry you can assume 'x' and 'z' to be divisible by 'd' and prove that 'y' too will be divisible by 'd' and
'y' and 'z' to be divisivle by 'd' and prove that 'x' too will be divisible by 'd';
hence all 3 cases will be completed.
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