Finding the Smallest 2-Digit Number Divisible by 2, 4, 6, 8, 10, and 12
Divisibility is a fundamental concept in mathematics, and understanding how to find the smallest 2-digit number divisible by multiple numbers can be incredibly useful for a variety of applications, from competitive problem-solving to real-world scenarios. In this article, we will explore techniques to determine the smallest 2-digit number that is divisible by 2, 4, 6, 8, 10, and 12. We will use the least common multiple (LCM) as our primary tool.
The LCM of 2, 4, 6, 8, 10, and 12
Let’s begin by finding the least common multiple (LCM) of these numbers. The LCM is the smallest number that is a multiple of each of the given numbers. While the original post states that the LCM is 853, this is incorrect. We will find the correct LCM step by step.
Step-by-Step Calculation of the LCM
The prime factorization of each number is as follows:
2 2 4 22 6 2 x 3 8 23 10 2 x 5 12 22 x 3To find the LCM, we take the highest power of each prime factor appearing in the factorizations:
23 3 5Thus, the LCM is 23 x 3 x 5 120.
Identifying 2-Digit Numbers Divisible by 2, 4, 6, 8, 10, and 12
Since 120 is a 3-digit number, we need to look for the smallest 2-digit number that is divisible by all these numbers. We can use the LCM to find this number.
Using the LCM to Find 2-Digit Numbers
The LCM of 2, 4, 6, 8, 10, and 12 is 120. Therefore, the smallest 2-digit number divisible by all these numbers is a factor of 120 and is determined by 120 ÷ 120 1, which is not a 2-digit number. Instead, we need to find multiples of the LCM that result in 2-digit numbers.
Listing 2-Digit Divisibles
The multiples of 120 that are 2-digit numbers are:
120 ÷ 5 24 (24 is the smallest 2-digit number divisible by 2, 4, 6, 8, 10, and 12) 120 ÷ 4 48 120 ÷ 3 72 120 ÷ 2 96Thus, the 2-digit numbers divisible by 2, 4, 6, 8, 10, and 12 are 24, 48, 72, and 96.
Alternative Approach Using Prime Factorization
An alternative approach involves prime factorization of each number and finding the least common multiple (LCM) of the factors.
Prime Factorization Method
Let's list the prime factors of each number:
2 2 4 22 6 2 x 3 8 23 10 2 x 5 12 22 x 3The LCM is the product of the highest power of each prime factor:
23 3 5Thus, the LCM is 120. Now, we find the 2-digit multiples of 120:
120 ÷ 5 24 120 ÷ 4 48 120 ÷ 3 72 120 ÷ 2 96So, the 2-digit numbers that are divisible by 2, 4, 6, 8, 10, and 12 are 24, 48, 72, and 96.