To find the smallest 3-digit number divisible by 16, 24, and 30, we need to follow these steps:
1. Find the least common multiple (LCM) of 16, 24, and 30.
- First, we calculate the LCM of 16 and 24.
- The prime factorization of 16 is [tex]\(2^4\)[/tex].
- The prime factorization of 24 is [tex]\(2^3 \times 3\)[/tex].
- To find the LCM, we take the highest power of each prime that appears in these factorizations:
- For the prime 2, we take [tex]\(2^4\)[/tex] because it is the highest power of 2 present.
- For the prime 3, we take [tex]\(3\)[/tex] because it is the highest power of 3 present.
- Thus, the LCM of 16 and 24 is [tex]\(2^4 \times 3 = 16 \times 3 = 48\)[/tex].
- Next, we calculate the LCM of the result (48) and 30.
- The prime factorization of 48 is [tex]\(2^4 \times 3\)[/tex].
- The prime factorization of 30 is [tex]\(2 \times 3 \times 5\)[/tex].
- To find the LCM, we take the highest power of each prime that appears in these factorizations:
- For the prime 2, we take [tex]\(2^4\)[/tex].
- For the prime 3, we take [tex]\(3\)[/tex].
- For the prime 5, we take [tex]\(5\)[/tex].
- Thus, the LCM of 48 and 30 is [tex]\(2^4 \times 3 \times 5 = 16 \times 3 \times 5 = 16 \times 15 = 240\)[/tex].
2. Find the smallest 3-digit number divisible by 240.
- The smallest 3-digit number is 100.
- We need to find the smallest number greater than or equal to 100 that is divisible by 240.
- We start from 100 and check each successive number to see if it is divisible by 240. However, it is quicker to find the smallest multiple:
- The smallest 3-digit multiple of 240 can be found by dividing 100 by 240 and then taking the ceiling of the quotient to find the next integer. The quotient of 100 divided by 240 is approximately 0.4167. The next integer is 1.
- Multiply 240 by this integer: [tex]\(240 \times 1 = 240\)[/tex].
Thus, the smallest 3-digit number divisible by 16, 24, and 30 is 240.
