How many numbers less than 1000 are multiples of both 10 and 13?

Options

A. 9
B. 8
C. 6
D. 7

Answer

To find how many numbers less than 1000 are multiples of both 10 and 13, we can follow these steps:

### Step 1: Find the LCM of 10 and 13
To find the least common multiple (LCM) of two numbers, we can use the formula:
\[ \text{LCM}(a, b) = \frac{a \times b}{\text{GCD}(a, b)} \]
Since 10 and 13 are coprime (they have no common factors other than 1), their GCD is 1. Therefore, the LCM is:
\[ \text{LCM}(10, 13) = \frac{10 \times 13}{1} = 130 \]

### Step 2: Determine how many multiples of 130 are less than 1000
Next, we need to find how many multiples of 130 are less than 1000. We can do this by dividing 1000 by 130:
\[ \frac{1000}{130} \approx 7.692 \]
Since we are looking for whole multiples, we take the integer part of this quotient, which is 7.

### Step 3: Conclusion
Thus, there are 7 multiples of both 10 and 13 that are less than 1000.

### Final Answer
The number of integers less than 1000 that are multiples of both 10 and 13 is **7**.

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