If a and b are integers such that a^2− b^2 = 100, what is the greatest possible value of a?
the greatest value of "a" corresponds with the least value of b^2
... zero is the integer with the smallest square
To find the greatest possible value of a, we need to understand the given equation a² - b² = 100.
We can see that this equation resembles the difference of squares formula: a² - b² = (a + b)(a - b).
So, we can rewrite the equation as (a + b)(a - b) = 100.
Now, we need to find two factors of 100 whose difference is even. This is because (a + b) and (a - b) are two factors of 100, and their difference needs to be even for a to be an integer.
The factors of 100 are:
1, 2, 4, 5, 10, 20, 25, 50, 100.
Now, let's try some pairs of factors and see if their difference is even:
(1, 100): Difference = 99 (Odd)
(2, 50): Difference = 48 (Even)
(4, 25): Difference = 21 (Odd)
(5, 20): Difference = 15 (Odd)
(10, 10): Difference = 0 (Even)
From the above pairs, we see that the only pair with an even difference is (10, 10).
Now, let's assign the values of (a + b) and (a - b) to be 10 each:
(a + b) = 10
(a - b) = 10
Adding these two equations, we get:
2a = 20
a = 10
Therefore, the greatest possible value of a is 10.
arrange to a^2 = b^2 + 10^2
so a must be the hypotenuse of a right-angled triangle
look at a list of small Pythagorean triples , and look at the hypotenuse:
leg1 leg2 hypotenuse
3 4 5
5 12 13
11 60 61
19 180 181
29 420 421
any multiples of those would also work.
Consider : 3, 4, 5
we also have 6, 8, 10 ----> 10^2 = 6^2 + 8^2,
so a=10, b=8
a = 10 is the largest and only value of a which satisfies your equation.
btw, did you know you can generate Pythagorean triples if
for any m and n, m > n , and m and n have no common factor between them, then
2mn, m^2 - n^2, and m^2 + n^2 will produce a Pythagorean triple.
e.g let m= 7 and n = 3
2mn = 42
m^2 - n^2 = 40
m^2 + n^2 = 58 and 42^2 + 40^2 = 58^2
If one of m and n is even , and the other odd, you will get a UNIQUE Pythagorean triple
e.g. m = 9, n = 4
2mn =72
m^2 - n^2 = 65
m^2 + n^2 = 97 -----> 97^2 = 72^2 + 65^2