Question

Consider the smallest five digit number $\text{\_6789}$. Guess the digit that can fit in the blank space, if the number is divisible by 9 in one case and 3 in the other case.

• A. 6 and 0, respectively.
• B. 3 and 6, respectively.
• C. 6 and 3, respectively.
• D. None of the above.

Answer 1

The correct option is C 6 and 3, respectively.

Let's assume the digit in black space be $d$.

For divisibility by 9:

Sum of all the digit must be divisible by 9.

$\Rightarrow d+6+7+8+9=d+30$

The smallest digit $d$ that can fit is 6 as

$\Rightarrow 6+30=36$

which is divisible by 9.

$\therefore d=6$

For divisibility by 3:

Sum of all the digit must be divisible by 3.

$\Rightarrow d+6+7+8+9=d+30$

Let the smallest digit $d=0$.

$\to 0+30=30$,

which is divisible by 3.

The number becomes $06789$, which is a four-digit number.

$\therefore d$ should be greater than 0.

For $d=3$,

Sum of all the digits = $3+6+7+8+9=33$, which is divisible by 3.

$\therefore d=3$

Hence, the digits are $6$ and $3$, respectively.