The greatest number less than 1500, which is divisible by both 16 and 18, is

Options

A. a) 1440
B. b) 1404
C. c) 1386
D. d) 1368

Answer

To find the greatest number less than 1500 that is divisible by both 16 and 18, we will follow these steps:

### Step 1: Find the Least Common Multiple (LCM) of 16 and 18.
To find the LCM, we can use the prime factorization method.

- The prime factorization of 16 is \(2^4\).
- The prime factorization of 18 is \(2^1 \times 3^2\).

Now, we take the highest power of each prime factor:
- For \(2\), the highest power is \(2^4\).
- For \(3\), the highest power is \(3^2\).

Thus, the LCM is:
\[ \text{LCM} = 2^4 \times 3^2 = 16 \times 9 = 144. \]

### Step 2: Find the greatest multiple of 144 that is less than 1500.
To find this, we divide 1500 by 144 and take the integer part of the result:
\[ \frac{1500}{144} \approx 10.4167. \]
Taking the integer part, we get 10.

### Step 3: Multiply the integer part by 144 to find the greatest multiple.
Now, we multiply 10 by 144:
\[ 10 \times 144 = 1440. \]

### Conclusion:
The greatest number less than 1500 that is divisible by both 16 and 18 is **1440**.

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