**Finding the greatest 4-digit number divisible by 48, 40, and 60**

To find the greatest 4-digit number that is divisible by 48, 40, and 60, we need to follow these steps:

**Step 1: Find the least common multiple (LCM) of the three numbers**
- LCM is the smallest number that is divisible by all the given numbers.
- Prime factorize each number and take the highest power of each prime factor:
- 48 = 2^4 * 3^1
- 40 = 2^3 * 5^1
- 60 = 2^2 * 3^1 * 5^1
- Multiply the highest powers of each prime factor:
- 2^4 * 3^1 * 5^1 = 2^4 * 3^1 * 5^1 = 2^4 * 3^1 * 5^1 = 2^4 * 3 * 5 = 2^4 * 15 = 720

Therefore, the LCM of 48, 40, and 60 is 720.

**Step 2: Find the greatest 4-digit number divisible by 720**
- The greatest 4-digit number is 9999.
- We need to find the largest multiple of 720 that is less than or equal to 9999.
- Divide 9999 by 720:
- 9999 ÷ 720 = 13 remainder 279
- Subtract the remainder from 9999:
- 9999 - 279 = 9720

Therefore, the greatest 4-digit number divisible by 720 is 9720.

**Step 3: Check if 9720 is divisible by 48, 40, and 60**
- Divide 9720 by 48:
- 9720 ÷ 48 = 202.5
- Since 202.5 is not a whole number, 9720 is not divisible by 48.
- Divide 9720 by 40:
- 9720 ÷ 40 = 243
- Since 243 is a whole number, 9720 is divisible by 40.
- Divide 9720 by 60:
- 9720 ÷ 60 = 162
- Since 162 is a whole number, 9720 is divisible by 60.

Therefore, the greatest 4-digit number divisible by 48, 40, and 60 is 9720.

In conclusion, the greatest 4-digit number that is exactly divisible by 48, 40, and 60 is 9720.

The solution is as follows:

1. First find the LCM of 40, 48, 60

LCM of 40, 48 and 60 : 2×2×2×2×3×5 = 240

2. Largest 4 digit No. is 9999

Divide 9999 by 240 i.e. 9999 ÷ 240
Quotient= 41 and Remainder=159

3. Now subtract 159 from 9999 i.e 9999 - 159 = 9840

Therefore 9840 is the greatest 4 digit number divisible by 40, 48 and 60.

We can check,
a. 9840÷ 40= 246
b. 9840÷48= 205
c. 9840÷60= 164

All the three quotients are whole numbers. Hence, we can conclude that 9840 is the greatest 4-digit number divisible by 40,48 and 60.

Hope you find this helpful :)