To determine how many positive integers between 100 and 700 (inclusive) are multiples of 7, follow these steps:

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Step 1: Identify the first and last multiples of 7 within the range.

• First multiple of 7 greater than or equal to 100: $\lceil \frac{100}{7} \rceil = \lceil 14.29 \rceil = 15$ So, $15 \times 7 = 105$.

• Last multiple of 7 less than or equal to 700: $\lfloor \frac{700}{7} \rfloor = \lfloor 100 \rfloor = 100$ So, $100 \times 7 = 700$.

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Step 2: Count the number of terms in the sequence.

The multiples of 7 between 105 and 700 form an arithmetic sequence: $105, 112, 119, \ldots, 700$

• The first term ($a$) is 105.
• The last term ($l$) is 700.
• The common difference ($d$) is 7.

The formula for the $n$-th term of an arithmetic sequence is: $l = a + (n-1) \cdot d$

Rearranging to find $n$: $n = \frac{l - a}{d} + 1$

Substitute the values: $n = \frac{700 - 105}{7} + 1 = \frac{595}{7} + 1 = 85 + 1 = 86$

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Final Answer:

There are 86 positive integers between 100 and 700 that are multiples of 7.

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Would you like more details or have any questions?
Here are 5 related questions to expand your understanding:

1. How many multiples of 9 are there between 200 and 900?
2. What is the sum of all multiples of 7 between 100 and 700?
3. How many integers between 500 and 1000 are divisible by both 7 and 5?
4. How would the solution change if the range were 150 to 650?
5. What is the largest multiple of 7 less than 1000?

Tip: When solving problems with arithmetic sequences, always verify the boundaries carefully to ensure inclusivity!