If the three-digit numbers A 28,3 B 9 and 62 C, where A, B and C are integers between 0 and 9 ,

step 1 In problem 40 we have three digit numbers A to 8 3b 9 6 to C where A B c are integers between 0

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

If the three-digit numbers $A 28,3 B 9$ and $62 \mathrm{C}$, where $A$, $B$ and $C$ are integers between 0 and 9 , are divisible by a fixed integer $k$, then the determinant $\left|\begin{array}{ccc}A & 3 & 6 \\ 8 & 9 & C \\ 2 & B & 2\end{array}\right|$ is (A) divisible by $k$ (B) divisible by $k^{2}$ (C) divisible by $2 k$ (D) None of these

If the three-digit numbers $A 28,3 B 9$ and $62 \mathrm{C}$, where $A$, $B$ and $C$ are integers between 0 and 9 , are divisible by a fixed integer $k$, then the determinant $\left|\begin{array}{ccc}A & 3 & 6 \\ 8 & 9 & C \\ 2 & B & 2\end{array}\right|$ is
(A) divisible by $k$
(B) divisible by $k^{2}$
(C) divisible by $2 k$
(D) None of these

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Key Concepts

Place Value Representation

This concept involves expressing multi?digit numbers in terms of the digits' contributions based on their positions (e.g., hundreds, tens, units). In problems where digits are variables and the numbers are subject to divisibility constraints, it is crucial to systematically convert these numbers into expressions involving the digits multiplied by powers of 10.

Determinant Properties

This concept covers the key characteristics and operations of determinants in matrices, such as the linearity in rows and columns and the effect of common factors. Understanding these properties helps in analyzing how the divisibility condition imposed on the entries of the matrix affects the divisibility of the determinant itself.

Divisibility and Modular Arithmetic

This concept concerns the idea that if a number is divisible by a fixed integer, then any linear combination of numbers that are multiples of that integer is also divisible by that integer. In the context of the problem, it implies that operations such as addition, subtraction, and multiplication (which appear in the determinant expansion) preserve divisibility by k.

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