Introduction to Finding the Lowest Number with a Specific Remainder

In mathematics, especially in modular arithmetic, finding the lowest number that leaves a specific remainder upon division by certain divisors is a common task. This article explains the step-by-step process to find the lowest number that leaves a remainder of 4 when divided by both 9 and 12.

Understanding the Problem

Our objective is to identify the smallest positive integer n such that:

n equiv 4 (mod 9) n equiv 4 (mod 12)

Step-by-Step Solution

Defining the Conditions

According to the given conditions, n - 4 must be divisible by both 9 and 12. This can be mathematically expressed as:

n ≡ 4 mod 9

n ≡ 4 mod 12

If n - 4 is divisible by both 9 and 12, then the smallest such n can be written as:

n k·lcm(9, 12) 4

Calculating the Least Common Multiple (LCM)

To find the LCM of 9 and 12, we first perform their prime factorizations:

Prime factorization of 9: 3^2 Prime factorization of 12: 2^2 × 3^1

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

For 2: 2^2 For 3: 3^2

Thus, the LCM of 9 and 12 is:

lcm(9, 12) 2^2 × 3^2 4 × 9 36

Formulating the Equation

Using the LCM, the equation for n becomes:

n 36k 4

Finding the Smallest Positive n

Name: For k 0

n 36 × 0 4 4

Name: For k 1

n 36 × 1 4 40

The smallest positive number that satisfies the given conditions is 40. Let's verify this:

40 ÷ 9 4 remainder 4 40 ÷ 12 3 remainder 4

Both division results show that 40 leaves a remainder of 4 when divided by both 9 and 12, confirming that the smallest positive number is 40.

Conclusion

The method described above can be generalized to solve similar problems. By calculating the LCM of the divisors and adding the given remainder, we can determine the smallest number that satisfies the given conditions. This approach is widely applicable in various fields including computer science and cryptography.

For further exploration, you can apply this method to other sets of divisors and remainders to deepen your understanding of modular arithmetic and least common multiples.