Question

Find the greatest number less than 10,000 which is exactly divisible by 48, 60 and 65.

Hint:

Always verify your final answer by dividing it by the largest of the three given numbers to make sure it is divisible.
Here, \(9360 \div 65 = 144\), which is an integer.
This simple check ensures that your LCM calculation and division steps are correct!

Answer 1

Step 1: Understanding the Question:
The topic of this question is Real Numbers, specifically focusing on the Least Common Multiple (LCM).
Any number that is exactly divisible by 48, 60, and 65 must be a multiple of their Least Common Multiple (LCM).
Our objective is to find the greatest multiple of this LCM that is strictly less than 10,000.
We will find the LCM using prime factorization, divide 10,000 by this LCM, and subtract the remainder to find the required number.

Step 2: Key Formula or Approach:
- Find the prime factorization of 48, 60, and 65.
- Calculate the LCM by taking the highest power of each prime factor present in the factorizations.
- Use division to find the largest multiple of the LCM below 10,000:
\[ \text{Required Number} = 10,000 - \text{Remainder of } \left(\frac{10,000}{\text{LCM}}\right) \]

Step 3: Detailed Explanation:

• Find the prime factorization of each number:
- For 48:
\[ 48 = 16 \times 3 = 2^4 \times 3 \]
- For 60:
\[ 60 = 4 \times 3 \times 5 = 2^2 \times 3 \times 5 \]
- For 65:
\[ 65 = 5 \times 13 \]

• Determine the Least Common Multiple (LCM) of 48, 60, and 65:
Identify the highest powers of all prime factors:
- Highest power of 2 is \(2^4\)
- Highest power of 3 is \(3^1\)
- Highest power of 5 is \(5^1\)
- Highest power of 13 is \(13^1\)
Multiply these values to calculate the LCM:
\[ \text{LCM} = 2^4 \times 3^1 \times 5^1 \times 13^1 \]
\[ \text{LCM} = 16 \times 3 \times 5 \times 13 \]
\[ \text{LCM} = 240 \times 13 = 3120 \]

• Find the largest multiple of 3120 that is less than 10,000:
Divide 10,000 by 3120:
\[ 10,000 = 3120 \times 3 + 640 \]
The quotient is 3 and the remainder is 640.

• Subtract the remainder from 10,000 to find the required divisible number:
\[ \text{Greatest Divisible Number} = 10,000 - 640 \]
\[ \text{Greatest Divisible Number} = 9360 \]


Step 4: Final Answer:
The greatest number less than 10,000 which is exactly divisible by 48, 60, and 65 is 9360.