What is the Smallest 3-Digit Number Divisible by Both 4 and 6?

The Importance of Least Common Multiple (LCM)

To find the smallest 3-digit number that is divisible by both 4 and 6, it's crucial to understand the concept of the Least Common Multiple (LCM). The LCM of two numbers is the smallest positive integer that is divisible by both of them. Here's the step-by-step process to find such a number.

Step 1: Find the Prime Factorization of the Numbers

First, we need to determine the LCM of 4 and 6 by finding their prime factorizations. - The prime factorization of 4 is (2^2). - The prime factorization of 6 is (2 times 3).

Step 2: Determine the LCM

To find the LCM, we take the highest power of each prime that appears in the factorization of either number. - For 2, the highest power is (2^2). - For 3, the highest power is (3^1). Therefore, the LCM of 4 and 6 is (2^2 times 3 4 times 3 12).

Step 3: Identify the Smallest 3-Digit Number Divisible by 12

The smallest 3-digit number is 100. We need to find the smallest multiple of 12 that is greater than or equal to 100. We start by dividing 100 by 12. - (100 div 12 approx 8.33). - Rounding up gives us 9.

Now multiply 9 by 12 to find the smallest 3-digit number that is divisible by 12:

- (9 times 12 108)

Thus, the smallest 3-digit number that is divisible by both 4 and 6 is 108.

Alternative Methods

While the LCM method is straightforward, there are alternative approaches to find the answer. - One approach is to start at the smallest 3-digit number, 100, and check each subsequent number for divisibility by both 4 and 6. - Another method involves recognizing that a number divisible by both 4 and 6 must be divisible by their LCM, 12. Therefore, we need an even number that is divisible by 3 (as the sum of its digits must be divisible by 3). Testing the first few even numbers greater than 100: - 100 is not divisible by 3. - 102 is even and divisible by 3, but not by 4. - 104 is divisible by 4 but not by 3. - 108 is divisible by 4, by 3, and therefore by 12.

Thus, 108 is the smallest 3-digit number divisible by both 4 and 6.

Verification

To verify, we can check the divisibility of 108 by 4 and 6. - (4 times 27 108) - (6 times 18 108) Since 108 satisfies both conditions, the answer is 108.

Conclusion

We have discussed the step-by-step process to find the smallest 3-digit number divisible by 4 and 6, using the LCM and alternative methods. The smallest such number is 108. Understanding these techniques is crucial for solving similar problems involving least common multiples and divisibility.