Finding the Smallest Number Greater Than 1000 with Remainder 3

Finding the Smallest Number Greater Than 1000 with Remainder 3

Mathematics often challenges us with intriguing problems that require both understanding and application of various concepts. One such problem involves finding the smallest number greater than 1000 that, when divided by any of the numbers 6, 9, 12, 15, or 18, leaves a remainder of 3. This article will guide you through the solution step-by-step, ensuring that by the end, you understand the process and can apply similar methods to solve similar problems.

The Problem at Hand

The problem statement is as follows: Find the smallest integer greater than 1000 which, upon division by 6, 9, 12, 15, or 18, leaves a remainder of 3. To approach this, we will follow several steps:

Step 1: Finding the Least Common Multiple (LCM)

The first step is to determine the least common multiple (LCM) of the given numbers. The LCM of a set of numbers is the smallest positive integer that is divisible by each of the numbers. For the numbers 6, 9, 12, 15, and 18, we proceed with the following steps:

Prime factorize each number:
6 2 × 3 9 32 12 22 × 3 15 3 × 5 18 2 × 32

To find the LCM, we take the highest powers of all primes involved: 22 (from 12), 32 (from 9 and 18), and 5 (from 15). Thus, the LCM is:

LCM 22 × 32 × 5 4 × 9 × 5 180.

Step 2: Multiples of the LCM

Next, we need to find the smallest multiple of 180 that is greater than 1000. This can be done by dividing 1000 by 180 and rounding up to the nearest whole number.

1000 ÷ 180 ≈ 5.56 (rounded up to 6)

Therefore, the smallest multiple of 180 greater than 1000 is:

180 × 6 1080.

Step 3: Adjusting for the Remainder

The question requires that the number leaves a remainder of 3 when divided by 6, 9, 12, 15, or 18. Since 1080 is divisible by all these numbers, we need to add 3 to 1080 to meet the condition of leaving a remainder of 3.

1080 3 1083.

Thus, 1083 is the smallest number greater than 1000 that, when divided by 6, 9, 12, 15, or 18, leaves a remainder of 3.

Verification

To ensure the solution is correct, we can verify it by dividing 1083 by each of the given numbers and checking for the remainder:

1083 ÷ 6 ≈ 180 remainder 3 1083 ÷ 9 ≈ 120 remainder 3 1083 ÷ 12 ≈ 90 remainder 3 1083 ÷ 15 ≈ 72 remainder 3 1083 ÷ 18 ≈ 60 remainder 3

Each of these divisions confirms that the remainder is indeed 3.

Conclusion

The smallest number greater than 1000 that, when divided by 6, 9, 12, 15, or 18, leaves a remainder of 3 is 1083. This problem illustrates the utility of the LCM method and the importance of considering remainders in modular arithmetic. By understanding these concepts, you can tackle a wide range of problems in number theory and related areas of mathematics.

References

1. Khan Academy: Least Common Multiple – Least Common Multiple 2. Purplemath: Least Common Multiple and Greatest Common Factor – LCM and GCF 3. MathIsFun: Least Common Multiple – LCM Guide