Let a, b, c, and n be positive integers. If a + b + c = 19(97) and
a + n = b - n = c/n,
compute the value of a.
This question is number theory related
+23246 a + b + c = 19(97)
a + n = b - n ---> 2n = b - a ---> a = b - 2n
b - n = c/n ---> n(b - n) = c
Substituting: a + b + c = 19(97) ---> (b - 2n) + (b) + ( n(b - n ) ) = 19(97)
---> 2b - 2n + bn - n2 = 19(97)
2b + bn - 2n - n2 = 19(97)
b(2 + n) - n(2 + n) = 19(97)
(b - n)(2 + n) = 19(97)
Using the hint 19(97), let's try: 2 + n = 19 and b - n = 97
---> n = 17 and b - 17 = 97 ---> b = 114
a = b - 2n ---> a = 114 - 2(17) = 80
c = n(b - n) = 17(114 - 17) = 1649
a + b + c = 80 + 114 + 1649 = 1843 and 19(97) = 1843
