What is the greatest five-digit number that is completely divisible by 8, 15, 16, 21, and 5?

Steps to Solve

1. Calculate the LCM of the given numbers To find a number that is divisible by 8, 15, 16, 21, and 5, we need to calculate the Least Common Multiple (LCM) of these numbers.

• The prime factorization of the numbers is:

  • $8 = 2^3$
  • $15 = 3^1 \times 5^1$
  • $16 = 2^4$
  • $21 = 3^1 \times 7^1$
  • $5 = 5^1$

• The LCM is taken by selecting the highest power of each prime:

  • $2^4$ from 16
  • $3^1$ from 15 or 21
  • $5^1$ from 15 or 5
  • $7^1$ from 21

Thus, LCM is calculated as: $$ \text{LCM} = 2^4 \times 3^1 \times 5^1 \times 7^1 $$

2. Calculate the value of the LCM Now we compute the LCM: $$ \text{LCM} = 16 \times 3 \times 5 \times 7 $$

First, calculate $16 \times 3$: $$ 16 \times 3 = 48 $$

Then, calculate $48 \times 5$: $$ 48 \times 5 = 240 $$

Finally, calculate $240 \times 7$: $$ 240 \times 7 = 1680 $$

So, the LCM of 8, 15, 16, 21, and 5 is $1680$.

3. Find the largest five-digit number divisible by LCM The largest five-digit number is 99999. To find the largest number that is divisible by $1680$, we divide $99999$ by $1680$ and take the floor of the result: $$ 99999 \div 1680 \approx 59.52 $$

Taking the floor gives us $59$.

4. Multiply to get the largest five-digit number Now, multiply $1680$ by $59$: $$ 1680 \times 59 = 99120 $$

This means that $99120$ is the largest five-digit number that is divisible by 8, 15, 16, 21, and 5.