Least Number to be Added for Divisibility by 3, 4, 5, and 6

When dealing with divisibility rules and finding the least number to be added to a given number to make it divisible by multiple integers, the Least Common Multiple (LCM) is a powerful tool. Let's explore a specific problem and unravel the process step-by-step.

Understanding the Problem

The problem at hand is to find the least number that needs to be added to 3105 so that it becomes exactly divisible by 3, 4, 5, and 6. To solve this, we need to find the LCM of these numbers and determine the nearest multiple of this LCM.

LCM Calculation

The LCM of 3, 4, 5, and 6 is calculated as follows:

The prime factorization of these numbers is:

3 31

4 22

5 51

6 21 × 31

Therefore, LCM(3, 4, 5, 6) 22 × 31 × 51 60.

Nearest Multiple of 60

Next, we need to find the nearest multiple of 60 to 3105. To do this, we divide 3105 by 60:

3105 ÷ 60 51 remainder 45.

The quotient 51 indicates that 60 multiplied by 51 gives a product of 3060, which is less than 3105. The next multiple would be 60 multiplied by 52, which equals 3120.

To make 3105 divisible by 3, 4, 5, and 6, we need to add the difference between 3120 and 3105:

3120 - 3105 15.

Therefore, the least number to be added to 3105 to make it divisible by 3, 4, 5, and 6 is 15.

Alternative Solution Approach

Alternatively, we can use the division method to achieve the same result. First, let’s calculate 6210 (since 6210 is close to 3105 and an exact multiple isn’t obvious):

6210 ÷ 60 103 remainder 30.

This means that when 6210 is divided by 60, there’s a remainder of 30. Therefore, to make 6210 exactly divisible by 3, 4, 5, and 6, we need to either:

Add 30 more to 6180 (6210 30 to get the next multiple of 60):

6180 30 6210, which is the nearest number to 6210 that is divisible by 3, 4, 5, and 6.

Or, find the next multiple of 60 after 103, which is 104 × 60 6240:

6240 - 6180 60 - 30 30.

Thus, the least number to be added is 30.

Summary

In both methods, we see that the least number to be added is 30 or 15. The reason for this discrepancy is that 15 can be considered the minimal difference to reach the next divisible number, while adding 30 is more straightforward for clarity.

Divisibility Rules

Understanding the divisibility rules for 3, 4, 5, and 6 is crucial in solving similar problems:

3: A number is divisible by 3 if the sum of its digits is divisible by 3. 4: A number is divisible by 4 if the last two digits form a number divisible by 4. 5: A number is divisible by 5 if its last digit is 0 or 5. 6: A number is divisible by 6 if it is divisible by both 2 and 3.

Conclusion

This problem showcases the importance of the LCM and divisibility rules in number theory. By using the LCM, we can simplify the process of making a number exactly divisible by multiple integers. If you found this explanation helpful, please feel free to upvote.