Find a number closest to 10,000 and greater than it that is divisible by both 5 and 12

Answer

To find a number closest to 10,000 that is greater than 10,000 and divisible by both 5 and 12, we can follow these steps:

### Step 1: Find the LCM of 5 and 12
To determine a number that is divisible by both 5 and 12, we first need to find the Least Common Multiple (LCM) of these two numbers.

- The prime factorization of 5 is \(5^1\).
- The prime factorization of 12 is \(2^2 \times 3^1\).

To find the LCM, we take the highest power of each prime factor:
- For 2, the highest power is \(2^2\).
- For 3, the highest power is \(3^1\).
- For 5, the highest power is \(5^1\).

Thus, the LCM is:
\[ LCM = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60 \]

### Step 2: Divide 10,000 by 60
Next, we divide 10,000 by 60 to find how many times 60 fits into 10,000.

\[ 10,000 \div 60 \approx 166.67 \]
This means that 60 fits into 10,000 approximately 166 times.

### Step 3: Calculate the largest multiple of 60 less than 10,000
To find the largest multiple of 60 that is less than or equal to 10,000, we take the integer part of the division result and multiply it back by 60.

\[ 166 \times 60 = 9,960 \]

### Step 4: Find the next multiple of 60 greater than 10,000
Now, to find the next multiple of 60 that is greater than 10,000, we simply add 60 to the largest multiple we just found.

\[ 9,960 + 60 = 10,020 \]

### Conclusion
Thus, the number closest to 10,000 that is greater than 10,000 and divisible by both 5 and 12 is **10,020**.

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