To solve this problem, we need to follow these steps:
1. Determine the Least Common Multiple (LCM) of 40, 48, and 60:
- First, we find the LCM of 40 and 48.
- Then, we use the result to find the LCM of that number with 60.
2. Find the Greatest 3-digit Number Divisible by the LCM:
- A 3-digit number ranges from 100 to 999.
- We need to determine the highest number within this range that is divisible by the LCM.
Let's go through the steps in detail:
### Step 1: Finding the LCM
- LCM of 40 and 48:
To find this, we can use the formula for the LCM of two numbers:
[tex]\[
\text{LCM}(a, b) = \frac{|a \times b|}{\text{GCD}(a, b)}
\][/tex]
- The Greatest Common Divisor (GCD) of 40 and 48 is 8.
- Therefore,
[tex]\[
\text{LCM}(40, 48) = \frac{40 \times 48}{8} = 240
\][/tex]
- LCM of 240 (result from above) and 60:
- The GCD of 240 and 60 is 60.
- So,
[tex]\[
\text{LCM}(240, 60) = \frac{240 \times 60}{60} = 240
\][/tex]
Hence, the LCM of 40, 48, and 60 is 240.
### Step 2: Find the Greatest 3-digit Number Divisible by 240
- We need to find the highest 3-digit number divisible by 240. The largest 3-digit number is 999.
- We divide 999 by 240 and take the integer part of the quotient, then multiply it back by 240 to get the largest number within 3 digits that is divisible by 240.
[tex]\[
\left\lfloor \frac{999}{240} \right\rfloor = 4 \quad \text{(quotient)}
\][/tex]
[tex]\[
4 \times 240 = 960
\][/tex]
So, the greatest 3-digit number which is exactly divisible by 40, 48, and 60 is 960.
