Find The Least Counting Number Which When Divided By 5 Gives The Remainder Of 1 And When Divided By 12

Find The Least Counting Number Which When Divided By 5 Gives The Remainder Of 1 And When Divided By 12 is a classic problem in number theory that involves understanding remainders, divisibility, and the application of the Chinese Remainder Theorem. These types of problems are fundamental in developing logical reasoning and problem-solving skills, especially in competitive exams and mathematical puzzles. In this article, we will explore how to approach this problem systematically, understand the underlying concepts, and find the least number satisfying both conditions.

Understanding the Problem

Before diving into the solution, it’s essential to carefully analyze the problem statement.

Restating the Problem

We need to find the smallest counting number (positive integer) that meets the following criteria:

• When divided by 5, the remainder is 1.
• When divided by 12, the remainder is 1.

Mathematically, this can be written as:

• n ≡ 1 (mod 5)
• n ≡ 1 (mod 12)

The goal is to find the least such number n.

Importance of the Problem

This problem demonstrates the application of modular arithmetic, and solving it provides insight into how different divisibility conditions can be combined to find a common solution. Such problems are foundational in number theory, cryptography, and algorithms.

Breaking Down the Conditions

To solve for n, we need to understand what the conditions imply.

Condition 1: n ≡ 1 (mod 5)

This means n can be expressed as:

• n = 5k + 1, where k is an integer (k ≥ 0).

Condition 2: n ≡ 1 (mod 12)

Similarly, n can be written as:

• n = 12m + 1, where m is an integer (m ≥ 0).

The problem reduces to finding the smallest positive integer n that satisfies both these equations simultaneously.

Using the Concept of Congruences

The above conditions can be viewed as a system of simultaneous congruences:

• n ≡ 1 (mod 5)


• n ≡ 1 (mod 12)

Since both congruences are of the form n ≡ 1 (mod divisor), the problem simplifies to finding n such that:

• n ≡ 1 (mod 5)


• n ≡ 1 (mod 12)

which implies that n - 1 is divisible by both 5 and 12.

Reformulating the Problem

If n - 1 is divisible by both 5 and 12, then:

• n - 1 ≡ 0 (mod 5)
• n - 1 ≡ 0 (mod 12)

So, n - 1 is a common multiple of 5 and 12.

Finding the Least Common Multiple (LCM)

The key is to find the smallest number that is divisible by both 5 and 12, which is the least common multiple (LCM) of these two numbers.

Calculating the LCM of 5 and 12

• Prime factorization:
• 5 = 5
• 12 = 2² × 3
• LCM is found by taking the highest powers of all prime factors present:
• LCM = 2² × 3 × 5 = 4 × 3 × 5 = 60

Thus, the smallest positive number divisible by both 5 and 12 is 60.

Determining n

Since n - 1 is divisible by 60, the smallest positive n is:

• n = 60 + 1 = 61

This number satisfies both conditions:

• 61 divided by 5 gives a quotient of 12 and a remainder of 1.
• 61 divided by 12 gives a quotient of 5 and a remainder of 1.

Therefore, the least counting number satisfying both conditions is 61.

Summary of the Solution

To summarize:

• Recognize that both conditions involve remainders of 1.
• Express the conditions as congruences:
• n ≡ 1 (mod 5)
• n ≡ 1 (mod 12)
• Transform the problem into finding the LCM of 5 and 12 to determine the smallest number where n - 1 is divisible by both.
• Calculate the LCM (which is 60).
• Add 1 to find the least number: 61.

General Approach to Similar Problems

This type of problem can be generalized to find the smallest number satisfying multiple modular conditions:

• Write each condition as a congruence.
• Identify the common structure, often involving the difference n - a multiple of each divisor.
• Calculate the least common multiple of the divisors involved.
• Determine n by adding the common remainder or offset to the LCM.

Example Variations

• Find the least number that leaves a remainder of 2 when divided by 3 and 4.
• Find the least number divisible by 6 and 8 with specific remainders.

Applying the same principles, you can solve these problems efficiently.

Practical Applications

Understanding such problems isn't just theoretical; they have practical applications:

• Cryptography: Modular arithmetic forms the backbone of encryption algorithms.
• Scheduling and Calendars: Determining cycles or repeating patterns involves solving similar congruences.
• Computer Science: Algorithms involving hashing, data synchronization, and distributed systems often rely on modular arithmetic.

Conclusion

In conclusion, finding the least counting number that meets specific divisibility and remainder conditions involves understanding modular arithmetic, calculating the least common multiple, and applying logical reasoning to combine the conditions. For the specific problem of finding the least number which, when divided by 5, gives a remainder of 1, and when divided by 12, gives a remainder of 1, the solution is 61. This approach can be extended to a wide range of similar problems, making it a vital skill in mathematical problem-solving and real-world applications.

Remember: Always analyze the problem carefully, express the conditions mathematically, and use the principles of LCM and congruences to find the solution efficiently.