Square Number Pairs from 1-50: Counting Rules

• MHB
• cooltu
In summary, given two integers from 1 to 50, where at least one is a square number and their sum is also a square number, the number of different pairs that can be found is being asked. The question also clarifies whether (9, 16) and (16, 9) should be counted as one pair. The equations $x^2+y=z^2$ and $x^2-z^2=(x-z)(x+z)=y$ are discussed, with the possibility of factoring y into mn and x-z into m and x+z into n. From this, it is concluded that x=(m+n)/2 and z=(n-m)/2, with a previous sign error being corrected.
cooltu
Two integers will be taken from 1 to 50, where at least one of them should be a square number and sum of them should also be a square number. How many different pair like this can be found? Will I count (9,16) and (16,9) as one ?

So $x^2+ y= z^2$ for x, y, and z integers. That is the same as $x^2- z^2= (x- z)(x+ z)= y$. Look at the ways to factor y: y= mn and the x- z= m, x+ z= n. Adding those two equations, 2x= m+ n, x= (m+ n)/2. Subtracting, 2z= n- m, z= (n- m)/2.

added much later: I've noticed that I have a sign error: from $x^2+ y= z^2$, $y= z^2- x^2$, not $x^2- x^2$. So y= (z- x)(z+ x). Taking y= mn, z- x= m, z+ x= n so that 2z= n+m, z= (n+m)/2, 2x= n- m so x= (n-m)/2, just the opposite of what I had before.

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1. What are square numbers?

Square numbers are numbers that result from multiplying a number by itself. For example, 4 is a square number because it is the result of 2 multiplied by itself (2 x 2 = 4).

2. How do you find square number pairs?

To find square number pairs, you can use the counting rule that states that the square of any number is found by adding the number to itself the same number of times as the number itself. For example, to find the square number pairs from 1-50, you would add 1 to itself once, 2 to itself twice, 3 to itself three times, and so on until you reach 50. The resulting numbers are the square number pairs.

3. What are the counting rules for square number pairs?

The counting rule for square number pairs is to add the number to itself the same number of times as the number itself. For example, to find the square number pairs from 1-50, you would add 1 to itself once, 2 to itself twice, 3 to itself three times, and so on until you reach 50.

4. How many square number pairs are there from 1-50?

There are 25 square number pairs from 1-50. This is because the highest square number that is less than or equal to 50 is 7 (7 x 7 = 49), and there are 7 numbers from 1-7 that are square numbers.

5. Why are square numbers important?

Square numbers are important in mathematics and science because they have many real-life applications. They are used in geometry to calculate the area of squares and other shapes, in physics to calculate distances and velocities, and in computer science to calculate memory and storage capacities. They also help in understanding patterns and relationships between numbers.

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