Finding the Smallest Number Which Leaves No Remainder and a Specific Remainder

In this article, we explore the mathematical problem of finding the smallest number that, when divided by several given denominators, leaves no remainder for some and a specific remainder for others. This involves understanding the principles of the Least Common Multiple (LCM) and modular arithmetic. Let's break down the solution step-by-step.

Solution to a Special Case

We are given four numbers: 12, 26, 44, and 72. We need to find a number that leaves no remainder when divided by 12, 26, 44, and 72, and leaves a remainder of 7 when divided by each of these numbers. This requires us to understand both the concept of LCM and the properties of remainders in modular arithmetic.

Solution for No Remainder

Step 1: Calculate the LCM of the denominators. - 12 22 × 3 - 26 2 × 13 - 44 22 × 11 - 72 23 × 32

The LCM is determined by taking the highest power of each prime factor involved:

- LCM 23 × 32 × 11 × 13 8 × 9 × 11 × 13 10296

Therefore, the smallest number that leaves no remainder when divided by 12, 26, 44, and 72 is:

- Smallest number LCM 10296

Step 2: Verify the solution. - 10296 ÷ 12 860 (remainder 0) - 10296 ÷ 26 395 (remainder 0) - 10296 ÷ 44 234 (remainder 0) - 10296 ÷ 72 143 (remainder 0)

Hence, 10296 is indeed the smallest number that leaves no remainder when divided by the given denominators.

Solution for Remainder 7

Step 1: Determine the smallest number that leaves a remainder of 7 when divided by the given denominators.

The smallest number that leaves a specific remainder r when divided by a set of divisors is the LCM of those divisors plus the remainder. Therefore, the number is:

- Smallest number LCM remainder

Step 2: Apply the formula. - Smallest number 10296 7 10303

Step 3: Verify the solution. - 10303 ÷ 12 858 (remainder 7) - 10303 ÷ 26 396 (remainder 7) - 10303 ÷ 44 234 (remainder 7) - 10303 ÷ 72 143 (remainder 7)

Hence, 10303 is the smallest number that leaves a remainder of 7 when divided by the given denominators.

Conclusion

By applying the principles of LCM and modular arithmetic, we can determine the smallest number that meets the specified criteria. This solution not only provides a clear step-by-step process but also demonstrates the mathematical reasoning behind it. Understanding these concepts is crucial in various fields, including computer science, engineering, and mathematics.

For further exploration, you might want to explore similar problems involving different divisors or remainders. This topic is fundamental and quite common in competitive programming and mathematical problem-solving scenarios.

Related Keywords

- Least Common Multiple (LCM) - Modular Arithmetic - Remainder Calculation

Keywords: Least Common Multiple, Modular Arithmetic, Remainder Calculation