Question

If a seven-digit number made up of all distinct digits $8,7,6,4,2,x,$ and $y$ is divisible by $3,$ then

• A. Maximum value of $x-y$ is $9$
• B. Maximum value of $x+y$ is $12$
• C. Minimum value of $xy$ is $0$
• D. Minimum value of $x+y$ is $3$.

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A Maximum value of $x-y$ is $9$
B Maximum value of $x+y$ is $12$
C Minimum value of $xy$ is $0$
D Minimum value of $x+y$ is $3$.

Note that the number is divisible by 3 if and only if the sum of its digits is divisible by 3.
Hence, the seven digit numbers formed by digits $2,4,6,7,8,x,y$ is divisible by 3 if and only if $2+4+6+7+8+x+y=27+x+y$ is divisible by 3 or $x+y$ is divisible by 3.
Now, $x$ and $y$ are chose from the set $\{0,1,3,5,9\}$. Hence the possible distinct pairs for $\left(x,y\right)$ are from the set $\left\{\left(0,3\right);\left(0,9\right);\left(1,5\right);\left(3,0\right);\left(3,9\right);\left(5,1\right);\left(9,0\right);\left(9,3\right)\right\}$
Therefore
A) The maximum value of $x-y$ is $9-0=9$
B) The maximum value of $x+y$ is $9+3=12$
C) The minimum value of $x\times y$ is $3\times 0=0$
D) The minimum value of $x+y$ is $3+0=3$