To determine the smallest number greater than 100,000 that is exactly divisible by each of 8, 15, and 21, we can follow these steps:
1. Identify the least common multiple (LCM) of the given numbers:
- The numbers we are dealing with are 8, 15, and 21.
- To find the LCM, we first find the prime factorizations of these numbers:
- [tex]\(8 = 2^3\)[/tex]
- [tex]\(15 = 3 \cdot 5\)[/tex]
- [tex]\(21 = 3 \cdot 7\)[/tex]
- The LCM is found by taking the highest power of each prime number that appears in the factorizations:
- For [tex]\(2\)[/tex], the highest power is [tex]\(2^3\)[/tex].
- For [tex]\(3\)[/tex], the highest power is [tex]\(3^1\)[/tex].
- For [tex]\(5\)[/tex], the highest power is [tex]\(5^1\)[/tex].
- For [tex]\(7\)[/tex], the highest power is [tex]\(7^1\)[/tex].
- Multiply these together to get the LCM:
[tex]\[
LCM = 2^3 \cdot 3^1 \cdot 5^1 \cdot 7^1 = 8 \cdot 3 \cdot 5 \cdot 7
\][/tex]
[tex]\[
LCM = 8 \cdot 3 = 24
\][/tex]
[tex]\[
24 \cdot 5 = 120
\][/tex]
[tex]\[
120 \cdot 7 = 840
\][/tex]
- Therefore, the least common multiple of 8, 15, and 21 is 840.
2. Find the number greater than 100,000 which is divisible by 840:
- To find the smallest number greater than 100,000 that is divisible by 840, we divide 100,000 by 840 and take the ceiling of the result, then multiply back by 840.
- Perform the division:
[tex]\[
\frac{100,000}{840} \approx 119.0476
\][/tex]
- Take the ceiling of [tex]\(119.0476\)[/tex], which is 120.
- Multiply back by 840 to find the number:
[tex]\[
120 \cdot 840 = 100,800
\][/tex]
So, the least number greater than 100,000 that is exactly divisible by 8, 15, and 21 is 100,800.
