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

3289

811

4078

735

✅ Correct Option: 2

Let the 4-digit number be $abcd$ , where $a$ is the thousands digit, $b$ is the hundreds digit, $c$ is the tens digit, and $d$ is the units digit.

From the given conditions:

$a + b + c = 15$ --- (1)

$b + c + d = 16$ --- (2)

$c = d + 6$ --- (3)

Since $c = d + 6$ and $c \leq 9$ , we get $d \leq 3$ , so $d$ can be $0, 1, 2,$ or $3$ .

Substituting $c = d + 6$ in Equation (2):

$b + (d + 6) + d = 16$

$b = 10 - 2d$

Substituting $b = 10 - 2d$ and $c = d + 6$ in Equation (1):

$a + (10 - 2d) + (d + 6) = 15$

$a = d - 1$

Testing all possible values of $d$ :

$d$	$a = d - 1$	$b = 10 - 2d$	$c = d + 6$	Number	Valid?
0	-1	10	6	-	No, $a$ is negative and $b$ is not a single digit
1	0	8	7	0871	No, $a = 0$ makes it a 3-digit number
2	1	6	8	1682	Yes
3	2	4	9	2493	Yes

Since $d$ can only range from 0 to 3, the only valid numbers are $1682$ and $2493$ .

Difference $= 2493 - 1682 = 811$

CAT 2021 Slot 2

For a $4$ -digit number, the sum of its digits in the thousands, hundreds and tens places is $14$ , the sum of its digits in the hundreds, tens and units places is $15$ , and the tens place digit is $4$ more than the units place digit. Then the highest possible $4$ -digit number satisfying the above conditions is

2025 Slot 3

The sum of all the digits of the number $(10^{50}+10^{25}-123)$ , is

CAT 2023 Slot 1

Brishti went on an 8-hour trip in a car. Before the trip, the car had travelled a total of x km till then, where x is a whole number and is palindromic, i.e., $x$ remains unchanged when its digits are reversed. At the end of the trip, the car had traveled a total of $26862 \mathrm{~km}$ till then, this number again being palindromic. If Brishti never drove at more than $110 \mathrm{~km} / \mathrm{hr}$ , then the greatest possible average speed at which she drove during the trip, in km/hr, was

CAT 2018 Slot 1

The number of integers $x$ such that $0.25 < 2^x < 200$ , and $2^x + 2$ is perfectly divisible by either $3$ or $4$ , is

CAT 2022 Slot 2

For some natural number $n$ , assume that $(15000)!$ is divisible by $(n!)!$ . The largest possible value of $n$ is

CAT 2020 Slot 1

How many $3-$ digit numbers are there, for which the product of their digits is more than $2$ but less than $7?$

Solution

Related questions:

Keyboard Shortcuts

CAT Past Year Questions (Topic-Wise)