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

### Concept & Formula When a question asks for a number to be a multiple of *both* $X$ and $Y$, it implies the number must be a multiple of their Least Common Multiple (LCM). $$LCM(10, 13)$$ Since $10$ and $13$ share no common prime factors (they are co-prime), their LCM is simply their product: $10 \times 13 = 130$. ### Step-by-Step Solution * **Given:** We need to find the count of numbers less than $1000$ that are multiples of both $10$ and $13$. * **Step 1: Determine the base multiple.** Any number divisible by both $10$ and $13$ must be divisible by $130$. * **Step 2: Find the maximum multiple under 1000.** We want to find how many times $130$ goes into numbers less than $1000$. Divide $999$ by $130$: $$\frac{999}{130} \approx 7.68$$ * **Step 3: Interpret the integer quotient.** The integer part of the quotient represents the exact number of multiples starting from $1$ up to $999$. These numbers are: $130, 260, 390, 520, 650, 780, 910$. There are exactly $7$ numbers. ### Exam Strategy & Shortcut Simply evaluate $\lfloor \frac{1000}{130} \rfloor$. Since $13 \times 7 = 91$, $130 \times 7 = 910$, which is under $1000$. The next multiple, $130 \times 8 = 1040$, exceeds $1000$. Thus, $7$ is immediately the answer without listing them. ### Common Pitfall Some might try to find the numbers divisible by $10$, then those divisible by $13$, and use a complex Venn diagram overlap method. This takes too long. Recognizing that "multiple of both" strictly means "multiple of the LCM" bypasses the unnecessary steps entirely. ### Final Answer **Therefore, the correct answer is 7.**