To solve the problem of finding the greatest number less than 100000 that is divisible by the numbers 48, 60, 64, and 80, we'll follow these steps:
1. Find the Least Common Multiple (LCM) of the numbers:
- Start by calculating the LCM of any two of the numbers, say 48 and 60.
- Then, find the LCM of this result with the next number, 64.
- Finally, find the LCM of this result with the last number, 80.
2. Calculate the greatest number under 100000 that is divisible by the LCM obtained in Step 1.
### Step-by-Step Solution:
1. Calculate the LCM of 48 and 60:
The LCM of two numbers, a and b, is given by:
[tex]\[
\text{LCM}(a, b) = \frac{a \times b}{\text{GCD}(a, b)}
\][/tex]
where GCD is the Greatest Common Divisor.
For 48 and 60:
[tex]\[
\text{LCM}(48, 60) = \frac{48 \times 60}{\text{GCD}(48 , 60)}
\][/tex]
The GCD of 48 and 60 is 12, therefore:
[tex]\[
\text{LCM}(48, 60) = \frac{48 \times 60}{12} = 240
\][/tex]
2. Calculate the LCM of 240 and 64:
Using the formula again:
[tex]\[
\text{LCM}(240, 64) = \frac{240 \times 64}{\text{GCD}(240, 64)}
\][/tex]
The GCD of 240 and 64 is 16, so:
[tex]\[
\text{LCM}(240, 64) = \frac{240 \times 64}{16} = 960
\][/tex]
3. Calculate the LCM of 960 and 80:
Once again using the formula:
[tex]\[
\text{LCM}(960, 80) = \frac{960 \times 80}{\text{GCD}(960, 80)}
\][/tex]
The GCD of 960 and 80 is 80, so:
[tex]\[
\text{LCM}(960, 80) = \frac{960 \times 80}{80} = 960
\][/tex]
Thus, the LCM of 48, 60, 64, and 80 is 960.
4. Find the greatest number less than 100000 which is divisible by the LCM (960):
To find the largest multiple of 960 that is less than 100000, divide 100000 by 960 and take the integer part of the quotient:
[tex]\[
\text{Integer part of} \left(\frac{100000}{960}\right) \approx 104.16667 \implies 104
\][/tex]
Now, multiply this quotient by 960:
[tex]\[
104 \times 960 = 99840
\][/tex]
### Conclusion:
The greatest number less than 100000 that is exactly divisible by 48, 60, 64, and 80 is 99840.
