Number System Question 2 - CAT 2025
For a 4-digit number (greater than 1000), the sum of the digits in the thousands, hundreds, and tens places is 15. The 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:
Step-by-Step Explanation
Let the thousands, hundreds, tens, and units digits of the number be a, b, c, and d, respectively.
- Set up the digit equations:
- a + b + c = 15 --- (Equation 1)
- b + c + d = 16 --- (Equation 2)
Subtracting Equation 1 from Equation 2 isolates the relationship between the boundary digits:
(b + c + d) - (a + b + c) = 16 - 15 ⇒ d = a + 1
- The tens digit is 6 more than the units digit:
c = d + 6 = (a + 1) + 6 = a + 7 --- (Equation 3)
- Express the hundreds digit in terms of a:
Substitute Equation 3 back into Equation 1:
a + b + (a + 7) = 15 ⇒ 2a + b = 8 ⇒ b = 8 - 2a
- Apply single-digit boundaries (0 ≤ digit ≤ 9):
- Since a is the thousands digit, a ≥ 1.
- For c = a + 7 ≤ 9, we find a ≤ 2.
Therefore, the only valid integer values for a are 1 and 2.
- Calculate the extreme numbers and their difference:
- If a = 1: b = 6,c = 8,d = 2 ⇒ 1682
- If a = 2: b = 4,c = 9,d = 3 ⇒ 2493
Difference = 2493 - 1682 = 811
Correct Option: A