Counting Integers Between 700 and 1000 with a Digit Sum of 10

Understanding the number of integers between 700 and 1000 where the sum of the digits equals 10 is a classic problem in number theory. This article explores various methods to solve this problem, focusing on the systematic approach to digit sum analysis and the optimization of counting techniques.

Introduction to the Problem

The problem statement requires us to find all three-digit integers between 700 and 1000 for which the sum of the digits is 10. These integers can be represented as 7xy, where x and y are the ten's and unit's digits, respectively. The key to solving this problem lies in systematically analyzing the possible values of x and y that satisfy the equation 7 x y 10.

Methodology and Analysis

First, let's simplify the equation:

7 x y 10

This simplifies to:

x y 3

This means we need to find all pairs of non-negative integers (x, y) that add up to 3.

Cases Based on the First Digit

Let's analyze the cases based on the first digit (thousand's place) of the number:

When the first digit is 7: The sum of the other two digits (x and y) must equal 3. The possible pairs are (0, 3), (1, 2), (2, 1), and (3, 0). When the first digit is 8: The sum of the other two digits (x and y) must equal 2. The possible pairs are (0, 2), (1, 1), and (2, 0). When the first digit is 9: The sum of the other two digits (x and y) must equal 1. The possible pairs are (0, 1), and (1, 0).

Each of these pairs corresponds to a valid integer:

When the first digit is 7: 703 712 721 730 When the first digit is 8: 802 811 820 When the first digit is 9: 901 910

By listing all possible combinations, we conclude that there are 9 integers between 700 and 1000 where the sum of the digits equals 10.

Conclusion

Through a detailed analysis of the digit sum, we have determined that there are 9 integers between 700 and 1000 whose digits sum to 10. This method can be generalized to similar problems involving digit sums and constrained ranges. Understanding the systematic approach to solving such problems is valuable for both theoretical and practical applications in mathematics and computer science.