Question 53846: Hi guys! Some of you may already know me as aaaaaaaa, and I'm asking a question:
Solve the equation:

j and c are in N (natural numbers).
I know that , but the question is: Is there a way to solve this without trial-and-error and find out that g=50, c=7?
Answer by AnlytcPhil(1806) (Show Source):
You can put this solution on YOUR website! Hi guys! Some of you may already know me as
aaaaaaaa, and I'm asking a question:
Solve the equation:
j² - (7c)² = 99
j and c are in N (natural numbers).
I know that j² - (7c)² = (j+7c)(j-7c), but the
question is: Is there a way to solve this without
trial-and-error and find out that j=50, c=7?
j² - (7c)² = 99
(j + 7c)(j - 7c) = 99
For that factorization of 99, the two factors,
(j + 7c) and (j - 7c) differ by 14c,
a multiple of 14
The only factors of 99 are 1, 3, 9, 11, 33, 99
So there are only three ways to write 99 as the product
of a larger natural number times a smaller natural number,
namely these three ways
99×1, 33×3 and 11×9.
Of the three ways only the first are factors that differ
by a multiple of 14, namely 98, since 98 = 14×7, so
14c = 14×7 so
c = 7
and since j² - (7c)² = 99,
j² - (7×7)² = 99
j² - 49² = 99
j² - 2401 = 99
j² = 2500
j = 50
Edwin
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