Question

If a seven digit number formed with distinct digits 4, 6, 9, 5, 3, \( x \) and \( y \) is divisible by 3, then the number of such ordered pairs \( (x, y) \) is

• A. \( 7 \)
• B. \( 8 \)
• C. \( 9 \)
• D. \( 10 \)

Hint:

A number is divisible by 3 if and only if the sum of its digits is divisible by 3. First, find the sum of the given digits. Then, determine the pairs of the remaining distinct digits whose sum, when added to the initial sum, results in a multiple of 3. Remember to consider ordered pairs \( (x, y) \) and \( (y, x) \) when \( x \neq y \).

Answer 1

The Correct Option is B

The given digits are 4, 6, 9, 5, 3, \( x \), and \( y \).
These are distinct digits, and \( x, y \in \{0, 1, 2, 7, 8 \} \) since they must be distinct from the given five digits.
Also \( x \neq y \).
For a number to be divisible by 3, the sum of its digits must be divisible by 3.
Sum of the given five digits = \( 4 + 6 + 9 + 5 + 3 = 27 \).
The sum of all seven digits is \( 27 + x + y \).
For this sum to be divisible by 3, \( x + y \) must be divisible by 3.
The possible pairs of distinct digits \( (x, y) \) from \( \{0, 1, 2, 7, 8 \} \) such that \( x + y \) is divisible by 3 are: \begin{itemize} \item \( 0 + 3 \) (3 not available) \item \( 0 + 6 \) (6 not available) \item \( 1 + 2 = 3 \) \( \implies (1, 2), (2, 1) \) \item \( 1 + 5 \) (5 not available) \item \( 1 + 8 = 9 \) \( \implies (1, 8), (8, 1) \) \item \( 2 + 4 \) (4 not available) \item \( 2 + 7 = 9 \) \( \implies (2, 7), (7, 2) \) \item \( 7 + 8 = 15 \) \( \implies (7, 8), (8, 7) \) \item \( 0 + ? \) (0, 3, 6, 9 - 3, 6, 9 not available) \end{itemize} The pairs \( (x, y) \) from \( \{0, 1, 2, 7, 8 \} \) such that \( x + y \) is divisible by 3 are: \begin{itemize} \item \( (0, 3) \) - not possible \item \( (1, 2) \), sum = 3 \item \( (1, 8) \), sum = 9 \item \( (2, 1) \), sum = 3 \item \( (2, 7) \), sum = 9 \item \( (7, 2) \), sum = 9 \item \( (7, 8) \), sum = 15 \item \( (8, 1) \), sum = 9 \item \( (8, 7) \), sum = 15 \end{itemize} The ordered pairs \( (x, y) \) are: \( (1, 2), (2, 1), (1, 8), (8, 1), (2, 7), (7, 2), (7, 8), (8, 7) \).
There are 8 such ordered pairs.