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CAT 2025Slot 3QAQues & Sol

Number SystemsHard

Question

For a 4-digit number (greater than 1000), sum of the digits in the thousands, hundreds, and tens places is 15. Sum of the digits in the hundreds, tens, and units places is 16. Also, the digit in the tens place is 6 more than the digit in the units place. The difference between the largest and smallest possible value of the number is

Options

Solution

1. Concept Used

  • Topic: Number Systems – Digit-based Algebraic Constraints
  • Formula: $$\text{If } a+b+c = S_1 \text{ and } b+c+d = S_2, \text{ then } d - a = S_2 - S_1$$

2. Calculation

Let the four-digit number be represented as $\overline{abcd}$, where $a$ is the thousands digit, $b$ is the hundreds digit, $c$ is the tens digit, and $d$ is the units digit.

Setting up the equations from the given conditions:

Condition 1: $a + b + c = 15 \quad \cdots (1)$

Condition 2: $b + c + d = 16 \quad \cdots (2)$

Condition 3: $c = d + 6 \quad \cdots (3)$

Subtracting equation (1) from equation (2):

$$(b + c + d) - (a + b + c) = 16 - 15$$

$$d - a = 1 \implies d = a + 1$$

Substituting $d = a + 1$ into Condition 3:

$$c = (a + 1) + 6 = a + 7$$

Substituting $c = a + 7$ into equation (1) to find $b$:

$$a + b + (a + 7) = 15$$

$$2a + b + 7 = 15$$

$$b = 8 - 2a$$

So the number takes the form: $\underline{a} \quad \underline{8-2a} \quad \underline{a+7} \quad \underline{a+1}$

Applying digit validity constraints (each digit must satisfy $0 \leq \text{digit} \leq 9$, and $a \geq 1$ since it's a 4-digit number):

  • From $b = 8 - 2a \geq 0$: $a \leq 4$
  • From $c = a + 7 \leq 9$: $a \leq 2$
  • From $d = a + 1 \geq 0$: always satisfied for $a \geq 1$
  • Combined with $a \geq 1$: $a \in {1, 2}$

Finding the two valid numbers:

When $a = 1$: digits are $1,\ 6,\ 8,\ 2$ → Number = $1682$

When $a = 2$: digits are $2,\ 4,\ 9,\ 3$ → Number = $2493$

Verification for $1682$: $1+6+8 = 15\ ✓,\ 6+8+2 = 16\ ✓,\ 8 - 2 = 6\ ✓$

Verification for $2493$: $2+4+9 = 15\ ✓,\ 4+9+3 = 16\ ✓,\ 9 - 3 = 6\ ✓$

Difference between the largest and smallest possible values:

$2493 - 1682 = \boxed{811}$

3. Solution

Answer = Option A ✅

The difference between the largest and smallest possible values of the number is 811.