Math Problem Statement

Write the smallest and the greatest 4-digit number whose digit sum is 15.

Solution

To find the smallest and greatest 4-digit numbers whose digit sum is 15, we need to consider two different cases.

Case 1: Smallest 4-digit number with a digit sum of 15

  1. Start with the smallest 4-digit number, which is 1000.
  2. We distribute the sum of 15 across the digits of the number while keeping the number as small as possible (with the smallest digits at the left).
  3. Start from the leftmost digit:
    • The first digit must be at least 1 (to keep it a 4-digit number).
    • Assign the rest of the sum (15 - 1 = 14) to the remaining digits, prioritizing lower values for subsequent digits.
  4. Distributing the remaining sum:
    • First digit: 1
    • Second digit: 4
    • Third digit: 0
    • Fourth digit: 0

Thus, the smallest number is 1050.

Case 2: Greatest 4-digit number with a digit sum of 15

  1. Start with the largest 4-digit number, which is 9999.
  2. Distribute the sum of 15 across the digits to maximize the number (with the largest digits at the left).
  3. Start from the leftmost digit:
    • Assign as much of the sum as possible to the first digit (up to 9).
    • Use the remaining sum for the subsequent digits.
  4. Distributing the remaining sum:
    • First digit: 9
    • Second digit: 6
    • Third digit: 0
    • Fourth digit: 0

Thus, the greatest number is 9600.

Final Answer:

  • Smallest number: 1050
  • Greatest number: 9600

Would you like a detailed explanation of the digit distribution process?
Here are 5 related questions for deeper exploration:

  1. What is the smallest 3-digit number with a digit sum of 15?
  2. How would the solution change if the digit sum requirement were different?
  3. What are other 4-digit numbers that also have a digit sum of 15?
  4. How can we generalize this process for n-digit numbers?
  5. What is the digit sum of the smallest and largest 5-digit numbers?

Tip: When solving such problems, always start from the extreme ends (smallest or largest digits) to distribute the sum strategically!

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Math Problem Analysis

Mathematical Concepts

Number Theory
Digit Sum Calculation
Place Value

Formulas

Sum of digits formula: sum(digits) = target sum

Theorems

None

Suitable Grade Level

Grades 6-8