Question

If a, b, c are integers such that $|a - b|^{19} + |c-a|^{19} = 1$, find the value of $|c - a| + |a - b| + |b - c|$

• A. 2
• B. 3
• C. 1
• D. 4

Answer 1

Answer: (A)

2

Answer 2

Let $X = |a - b|$ and $Y = |c - a|$. Since $a, b, c$ are integers, $X$ and $Y$ are non-negative integers. The equation is $X^{19} + Y^{19} = 1$. The only integer solutions are $(X, Y) = (0, 1)$ or $(1, 0)$.

Case 1: $|a - b| = 0$ and $|c - a| = 1$. This implies $a = b$. The expression is $|c - a| + |a - b| + |b - c| = 1 + 0 + |a - c| = 1 + 0 + 1 = 2$.

Case 2: $|a - b| = 1$ and $|c - a| = 0$. This implies $c = a$. The expression is $|c - a| + |a - b| + |b - c| = 0 + 1 + |b - a| = 0 + 1 + 1 = 2$.

In both cases, the value is 2.