What is the Smallest Number Which When Divided by 15, 18, and 27 Leaves 2 as a Remainder in Each Case?

When faced with the problem of finding the smallest number that leaves a remainder of 2 when divided by 15, 18, and 27, the solution involves using the concept of the least common multiple (LCM). This article will guide you step-by-step through the process of finding such a number, explaining the underlying mathematical principles and providing examples.

Understanding the Problem

Initially, we start with the requirement: We are looking for the smallest number N such that when divided by 15, 18, and 27, the remainder is 2 in each case.

Step-by-Step Approach

Adjust the Divisors: Since we want the number to leave a remainder of 2, we can subtract 2 from N. This means we are looking for a number N such that N - 2 is divisible by 15, 18, and 27. Find the Least Common Multiple (LCM): We need to compute the LCM of the numbers 15, 18, and 27. This involves finding a number that is a multiple of all three numbers. Prime Factorization: Start by finding the prime factorization of each number. Compute the LCM: Take the highest power of each prime factor across the three numbers. For 2: (2^1) from 18, for 3: (3^3) from 27, and for 5: (5^1) from 15. The LCM is calculated as:

Prime Factorization

15 31 × 51
18 21 × 32
27 33

Compute the LCM

The LCM is calculated by:

LCM 21 × 33 × 51 2 × 27 × 5 270.

Add the Remainder Back

Since we are looking for N and we have determined that N - 2 is 270, we simply add the remainder 2 back to the LCM to get N:
N 270 2 272.

Conclusion and Additional Insights

The smallest number which, when divided by 15, 18, and 27, leaves a remainder of 2 in each case is 272.

To verify, we can check the division:

272 ÷ 15 18 remainder 2 272 ÷ 18 15 remainder 2 272 ÷ 27 10 remainder 2

This article has provided a comprehensive guide to solving a math problem using the concept of LCM and remainder. For further practice, you can try finding all the integers from 1 to 1000 that leave a remainder of 2 when divided by 15, 18, and 27. The numbers are: 2, 272, 542, and 812.

Related Topics and Resources

Explore other articles on mathematics, such as finding the LCM of different sets of numbers or other remainder problems. Understanding these concepts can enhance your problem-solving skills and prepare you for a wide range of mathematical challenges.