Let the three digit numbers A 28,3 B 9, and 62 C, where A,...

Let the three digit numbers $A 28,3 B 9$, and $62 C$, where $A, B$, and $C$ are integers between 0 and 9 , be divisible by a fixed integer $k$. Show that the determin $\left|\begin{array}{ccc}A & 3 & 6 \\ 8 & 9 & C \\ 2 & B & 2\end{array}\right|$ nt is divisible by $k .$

Key Concepts

Divisibility

Divisibility is a fundamental concept in number theory that describes when one integer can be evenly divided by another, meaning there is no remainder. In problems dealing with divisibility, it is common to show that certain arithmetic expressions or combinations, such as determinants or sums, inherit the divisibility properties of their component numbers.

Modular Arithmetic

Modular arithmetic, or arithmetic modulo an integer, is a system where numbers wrap around after reaching a certain value (the modulus). It allows one to work with remainders and is indispensable in proving statements about divisibility because it reduces complex expressions to simpler equivalence classes that preserve divisibility properties.

Place Value Representation

Place value representation is the concept that the value of a digit in a number is determined by its position (hundreds, tens, ones, etc.). This is particularly useful in translating three?digit numbers into algebraic expressions, thereby facilitating the analysis of their divisibility by treating the digits as variables multiplied by powers of 10.

Determinant of a Matrix

The determinant is a scalar value computed from a square matrix that encodes important properties of the matrix, such as invertibility. In problems that combine linear algebra with number theory, determinants can be analyzed under modular arithmetic to show that if the elements of a matrix satisfy certain divisibility conditions, then the determinant itself will also be divisible by the same integer.

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