If xi=ai br ci, i=1,2,3, are theree-digit positive integers...

If $x_{i}=a_{i} b_{r} c_{i}, i=1,2,3$, are theree-digit positive integers such that each $x_{i}$ is a multiple of 19 , then for some integer $n, \Delta=\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3}\end{array}\right|$ is given by (A) $19 n+1$ (B) $19 n+2$ (C) $19 n$ (D) $19 n+3$

Key Concepts

Divisibility by a Prime Number

This concept pertains to understanding when an integer is a multiple of a given prime number. In many problems, including those with constraints on digits or structure, confirming that certain expressions are divisible by a prime involves checking that they vanish modulo that prime. This approach can reveal underlying patterns and structural invariance that are key to solving more complex problems.

Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers, where numbers wrap around upon reaching a certain value—the modulus. In problems dealing with divisibility, one typically works modulo a prime to simplify computations and to understand how sums, differences, or products behave in relation to that prime. This method is essential for proofs and arguments involving congruences.

Determinants in Linear Algebra

A determinant is a scalar property computed from a square matrix, capturing essential features like volume scaling, orientation, and invertibility of the matrix. Its multilinear nature with respect to the rows (or columns) makes it sensitive to linear combinations, and it can often be expressed in forms that reveal modular properties when the matrix entries possess specific divisibility characteristics.

Congruence Invariance in Determinants

When matrix entries are defined so that certain products or sums are divisible by a fixed number, the determinant may inherit a modular form. This concept involves recognizing that the determinant, being a combination of the matrix elements through addition and multiplication, often adheres to congruence relations, allowing it to be written in terms of a fixed modulus plus a constant. Such insights are valuable when the problem requires expressing the determinant in a form like 'm*n + k'.

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