JEE Mathematics 2023 Question with Solution

Given: Three-digit numbers are to be formed using the digits $1, 3, 5, 8$ with repetition allowed, and the number must be divisible by $3$.

Find: The total number of such three-digit numbers.

Use the divisibility rule of $3$: a number is divisible by $3$ if the sum of its digits is divisible by $3$.

The given digits have remainders modulo $3$ as follows:

$$\begin{aligned} 1 &\equiv 1 \pmod{3} \\ 3 &\equiv 0 \pmod{3} \\ 5 &\equiv 2 \pmod{3} \\ 8 &\equiv 2 \pmod{3} \end{aligned}$$

So the available remainder classes are:

• remainder $0$: $\{3\}$
• remainder $1$: $\{1\}$
• remainder $2$: $\{5,8\}$

For a three-digit number, the sum of the three remainders must be $0 \pmod{3}$.

Possible remainder combinations are:

$$(0,0,0), \ (1,1,1), \ (2,2,2), \ (0,1,2)$$

Now count each case.

Case I: $(0,0,0)$ Only the digit $3$ is available.

$$\Rightarrow 1 \text{ number}$$

Case II: $(1,1,1)$ Only the digit $1$ is available.

$$\Rightarrow 1 \text{ number}$$

Case III: $(2,2,2)$ Each place can be filled by either $5$ or $8$.

$$2^3 = 8$$

So this case gives $8$ numbers.

Case IV: $(0,1,2)$ Digits can be chosen in

$$1 \times 1 \times 2 = 2$$

ways, and these can be arranged in

$$3! = 6$$

ways. Hence the total number in this case is

$$2 \times 6 = 12$$

Adding all valid cases:

$$1 + 1 + 8 + 12 = 22$$

Therefore, the total number of three-digit numbers is $22$, so the correct option is D.

Given: Digits are $1, 3, 5, 8$ and repetition is allowed.

Find: How many three-digit numbers formed are divisible by $3$.

A quick way is to classify digits by their remainders modulo $3$:

• $3$ gives remainder $0$
• $1$ gives remainder $1$
• $5, 8$ give remainder $2$

Then count only those three-position remainder patterns whose sum is divisible by $3$:

$$(0,0,0), \ (1,1,1), \ (2,2,2), \ (0,1,2)$$

Their counts are:

• $1$ for $(0,0,0)$
• $1$ for $(1,1,1)$
• $2^3 = 8$ for $(2,2,2)$
• $1 \times 1 \times 2 \times 3! = 12$ for $(0,1,2)$

So the total is

$$1 + 1 + 8 + 12 = 22$$

This works because divisibility by $3$ depends only on the sum of remainders, not on the exact digits themselves. Therefore, the correct option is D.