Answer:
3^2 + 11^2 = 130
7^2 + 9^2 = 130
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1^2 + 18^2 = 325
6^2 + 17^2 = 325
Explanation:
We want to express 130 and 325 as the sum of two squares in two different ways.
Then, we want to find two integers a and b such that:
a^2 + b^2 = 130
There is not an analytical way to do this, we just need to try with different integers.
Let's start with 130.
So we start by defining a as a really small integer, for example 1, and try to find b.
1^2 + b^2 = 130
b^2 = 130 - 1
b^2 = √129 = 11.3
This is not an integer, so let's try with another value of a.
a = 2
2^2 + b^2 = 130
4 + b^2 = 130
b^2 = 130 - 4
b = √126 = 11.2
This can be discarded again.
Now let's try with a = 3
3^2 + b^2 = 130
9 + b^2 = 130
b^2 = 130 - 9
b = √121 = 11
Nice.
So we can express 130 as:
3^2 + 11^2 = 130
Now let's find another pair.
In the same way, we can see that for a = 7 we get:
7^2 + b^2 = 130
42 + b^2 = 130
b^2 = 130 - 49
b = √81 = 9
Then we can write:
7^2 + 9^2 = 130
Now for 325:
With the same reasoning than before, we want to find two integers such that:
a^2 + b^2 = 325
Then we start evaluating a by the smallest values:
a = 1
1^2 + b^2 = 325
b^2 = 325 - 1 = 324
b = √324 = 18
Then we can write 325 as:
1^2 + 18^2 = 325
Now to find the next pair we need to keep testing values for a, we will get for a = 6:
6^2 + b^2 = 325
36 + b^2 = 325
b^2 = 325 - 36 = 289
b = √289 = 17
Then we can write:
6^2 + 17^2 = 325.