Finding the Least Number with Specific Remainders Using LCM

When solving the problem of finding the least number that, when divided by 8, 12, 18, and 24, leaves a remainder of 5 each time, we can follow a systematic approach involving the Least Common Multiple (LCM). This method ensures an efficient solution that aligns with the principles of number theory.

Understanding the Problem

The problem states that a number, when divided by 8, 12, 18, and 24, should leave a remainder of 5 in each case. This can be mathematically represented as:

Number % 8 5 Number % 12 5 Number % 18 5 Number % 24 5

To solve this, we first adjust each divisor by subtracting the remainder from it:

8 - 5 3 12 - 5 7 18 - 5 13 24 - 5 19

Calculating the LCM

The LCM of the adjusted divisors (3, 7, 13, 19) is the smallest positive integer that is divisible by each of these numbers. To find the LCM, we can break down each number into its prime factors:

3 3 7 7 13 13 19 19

Since each of these numbers are prime and do not share common factors, the LCM is simply the product of these numbers:

LCM 3 × 7 × 13 × 19

Calculating this:

3 × 7 21

21 × 13 273

273 × 19 5187

However, we need to realize that the problem can be simplified by finding the LCM of the original divisors (8, 12, 18, 24) and then adjusting accordingly. Let's use the original divisors to find the LCM:

Prime Factorization and Finding LCM

First, we perform the prime factorization of each divisor:

8 2^3 12 2^2 × 3^1 18 2^1 × 3^2 24 2^3 × 3^1

Next, we take the highest power of each prime factor:

For 2: The highest power is 2^3 from 8 and 24. For 3: The highest power is 3^2 from 18.

Therefore, the LCM is:

LCM 2^3 × 3^2 8 × 9 72

Once we have the LCM of the original divisors, we add the initial remainder (5) to this LCM:

72 5 77

Verification

To verify, let's check the remainder when 77 is divided by 8, 12, 18, and 24:

77 ÷ 8 9 remainder 5

77 ÷ 12 6 remainder 5

77 ÷ 18 4 remainder 5

77 ÷ 24 3 remainder 5

These results confirm that 77 is indeed the least number that leaves a remainder of 5 when divided by 8, 12, 18, and 24.

Conclusion

In conclusion, the least number when divided by 8, 12, 18, and 24 leaves a remainder of 5 in each case is 77. This solution leverages the concept of LCM and verifies the results through simple arithmetic, ensuring a correct and efficient method.

Additional Information

For those interested in a more efficient method, we can also use the division method to find the LCM of 8, 12, 18, and 24. This method involves dividing by the common factors until the remainder is 1. In this case, we find the LCM to be 72, and adding the remainder of 5 gives us 77.