Determine the Three-Digit Numbers Divisible by 37 with a Remainder of 9 and Calculate Their Sum

Introduction

In this article, we will embark on a mathematical journey to identify and calculate the sum of three-digit numbers that, when divided by 37, yield a remainder of 9. We will delve into the concepts of modular arithmetic and divisibility criteria to solve this intriguing problem.

Understanding Modular Arithmetic

Modular arithmetic is a branch of number theory that deals with operations on integers modulo a fixed positive integer. In our case, the modulus is 37. When we say a number x is divisible by 37 with a remainder of 9, we mean that x can be expressed as x = 37k + 9, where k is an integer.

Finding the Three-Digit Numbers

To find the three-digit numbers that satisfy the given condition, we can use the above modular equation. We need to find all values of x between 100 and 999 such that x = 37k + 9. Starting with k = 0, we get x = 37(0) + 9 = 9. This is a three-digit number with the desired remainder. Continuing with k = 1, we get x = 37(1) + 9 = 46. This is not a three-digit number. We can continue this process until we find all the three-digit numbers that meet the condition. The numbers are 9, 45, 81, 117, 153, 189, 225, 261, 297, 333, 369, 405, 441, 477, 513, 549, 585, 621, 657, 693, 729, 765, 801, 837, 873, 909, 945, and 981.

Calculating the Sum

Now that we have identified all the three-digit numbers that are divisible by 37 with a remainder of 9, we can calculate their sum. Sum = 9 + 45 + 81 + 117 + 153 + 189 + 225 + 261 + 297 + 333 + 369 + 405 + 441 + 477 + 513 + 549 + 585 + 621 + 657 + 693 + 729 + 765 + 801 + 837 + 873 + 909 + 945 + 981 = 13260

Conclusion

In conclusion, we have successfully identified all the three-digit numbers divisible by 37 with a remainder of 9, which are 9, 45, 81, 117, 153, 189, 225, 261, 297, 333, 369, 405, 441, 477, 513, 549, 585, 621, 657, 693, 729, 765, 801, 837, 873, 909, 945, and 981. We have also calculated their sum to be 13260. Understanding modular arithmetic and divisibility criteria is essential for solving problems like these. We hope this article has provided a clear and detailed explanation of the concepts and methods involved.

Determinați Câte Numere De Trei Cifre Împărțite La 37 Dau Restul 9 Și Calculați Suma Acestor Numere