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How many pairs of positive integer a, b are such that [tex]a^2 + b^2 = 121?[/tex]
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The number of pairs of positive integers a and b can be found by using the formula n(n+1)/2, where n is the largest number that both a and b can be. This formula is derived from the fact that for every value of a, there are n possible values of b, and vice versa. Therefore, the total number of pairs is the sum of all possible values for a and b, which is n + n + n... (n times) = n^2. However, since we are only looking at positive integers, we divide by 2 to get n(n+1)/2.
Ordered pairs are pairs of numbers where the order matters. For example, (1,2) and (2,1) are considered different ordered pairs. Unordered pairs, on the other hand, are pairs of numbers where the order does not matter. In the previous example, (1,2) and (2,1) would be considered the same unordered pair.
Sure, let's say we want to find the number of pairs of positive integers a and b, where a can be any number from 1 to 3, and b can be any number from 1 to 2. Using the formula n(n+1)/2, where n is the highest number that a and b can both be, we get (3)(3+1)/2 = 6. Therefore, there are 6 pairs of positive integers (a,b) that satisfy these conditions.
No, there is no limit to the number of pairs of positive integers a and b. The formula n(n+1)/2 can be used for any value of n, so as long as there is a positive integer limit for a and b, there will be a finite number of pairs.
No, the formula n(n+1)/2 can only be used for finding the number of pairs of positive integers a and b. For non-positive integers, a different formula would need to be used.