If a and b are integers such that a2 − b2 = 100, what is the greatest possible value of a?
typo, you meant
a^2 − b^2 = 100
discussed here:
https://www.jiskha.com/questions/1774797/If-a-and-b-are-integers-such-that-a-2-b-2-100-what-is-the-greatest-possible
To find the greatest possible value of "a", we can start by factoring the equation a^2 - b^2 = 100 using the difference of squares formula.
The difference of squares formula states that a^2 - b^2 = (a + b)(a - b).
So, we have (a + b)(a - b) = 100.
Now, let's consider the possible factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, and 100.
Since "a" and "b" are integers, both (a + b) and (a - b) must be integers.
Thus, we need to find a pair of factors of 100 such that their sum and difference are both integers.
If we try to find such pairs, we can see that (a + b) = 50 and (a - b) = 2 is a valid solution.
Solving these equations simultaneously, we get:
a + b = 50 ...(1)
a - b = 2 ...(2)
By adding equations (1) and (2), we get:
2a = 52
a = 26
So, the greatest possible value of "a" is 26.
To find the greatest possible value of a, we need to consider the given equation: a^2 - b^2 = 100.
Since we are looking for integers a and b, we can rephrase the equation by factoring the difference of squares: (a - b)(a + b) = 100.
Now, we need to factorize 100 into pairs of integers. The pairs of factors of 100 are: (1, 100), (2, 50), (4, 25), and (5, 20).
To determine the greatest possible value of a, we need to find the pair where the sum of the factors (a - b) and (a + b) is the largest.
For the pair (1, 100), the sum is 1 + 100 = 101.
For the pair (2, 50), the sum is 2 + 50 = 52.
For the pair (4, 25), the sum is 4 + 25 = 29.
For the pair (5, 20), the sum is 5 + 20 = 25.
Therefore, the greatest possible sum is 101, which corresponds to the pair (1, 100).
Since a + b = 101, we can solve for a by setting b = 100:
a + 100 = 101
a = 101 - 100
a = 1
Thus, the greatest possible value of a is 1.