plz help
1.for any 2 non-zero integers x and y there exist a unique integer q and r called the quotient and remainder respectively such that x=yq+r where <=r|y|
i.e
division algorithm
2.if m is a prime integer, show that m/xy implies m/x and m/y
sshow step please
Of course! I'm here to help. Let's go step by step to answer both of your questions:
1. Division Algorithm:
To prove the division algorithm, we need to show that for any two non-zero integers, x and y, there exist a unique integer quotient q and remainder r such that x = yq + r, where the absolute value of r is less than or equal to the absolute value of y.
Step 1: Existence of q and r:
To show that such integers q and r exist, consider the case when y is positive. We can start subtracting multiples of y from x until we reach a non-negative number r. Let q be the number of subtractions necessary. Mathematically, we can express this as:
x = yq + r, where r ≥ 0 and r < y.
Similarly, if y is negative, we can start adding multiples of y to x until we reach a non-negative number r. In this case, q would be negative, indicating the addition of y to x. Mathematically, we can express this as:
x = yq + r, where r ≥ 0 and r < |y|.
Step 2: Uniqueness of q and r:
To prove that q and r are unique, suppose there are two sets of integers q1, q2, r1, and r2 that satisfy the equation x = yq1 + r1 and x = yq2 + r2. In this case, we need to show that q1 = q2 and r1 = r2.
Firstly, subtracting the two equations, we get:
y(q1 - q2) = r1 - r2
Since y is non-zero, (q1 - q2) is also non-zero. This implies that the absolute value of r1 - r2 must be at least as large as the absolute value of y.
However, as per the division algorithm, the remainder r must be less than or equal to the absolute value of y. Therefore, r1 - r2 should be less than or equal to the absolute value of y. This means that r1 - r2 = 0, leading to r1 = r2.
Substituting r1 = r2 into the previous equation, we obtain:
y(q1 - q2) = 0
Dividing both sides by y, and knowing that y is non-zero, we conclude that q1 - q2 = 0, which implies q1 = q2.
Hence, we have proved the existence and uniqueness of q and r satisfying x = yq + r, where the absolute value of r is less than or equal to the absolute value of y.
2. The implication m/xy → (m/x and m/y):
To prove that if m is a prime integer, then m/xy implies m/x and m/y, we need to show that if m is divisible by xy, then it must be divisible by both x and y.
Step 1: Assume m/xy:
Let's assume that m is divisible by xy, which can be expressed as m = k(xy), where k is an integer.
Step 2: Rewrite m in terms of x and y:
We can rewrite m = k(xy) as m = (kx)y. By comparing this equation with the general form of the division algorithm x = yq + r, we can see that y divides m without any remainder, and q = kx.
Step 3: Prove that m is divisible by x:
Since q = kx, we can substitute q back into the equation m = (kx)y as m = qy. This implies that m is divisible by y, as per the division algorithm. Therefore, m is divisible by x.
Step 4: Prove that m is divisible by y:
We have already shown that m = qy, where q is an integer. Hence, m is clearly divisible by y.
Therefore, we have shown that if m is a prime integer and m/xy holds, then m/x and m/y also hold.
I hope this explanation helps you understand these concepts better. If you have any further questions, feel free to ask!