Sequences: Consecutive Integers - Counting Consecutive Multiples within a Range
How many multiples of 7 are there between 700 and 1000, inclusive?
If you're not clear on how to reach 994 as the highest multiple of 7 that is close to 1000, "jump" with known multiples of 7 until you reach as close to 1000 as possible: 700 is a multiple of 7, so 980 (which is $$700+280$$) is also a multiple of 7, which is closer to 1000.
Can we go closer? Sure! Add 14 (another multiple of 7) to 980 to get to 994. This is the closest multiple of 7.
Incorrect.
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Correct.
Remember the 3-step method for counting the number of multiples of $$x$$ within a range:
1. Find the relevant *extremes*—the nearest *multiples* of $$x$$ within the specified range:
>A number is divisible by 7 if
>>$$2\times$$ [number of hundreds] $$+$$ [remaining number]
>is divisible by 7. Thus, the least multiple of 7 in the set is 700 and the greatest is 994.
2. Subtract the *relevant extremes* and divide by $$x$$:
>>$$\displaystyle \frac{994-700}{7} = \frac{294}{7} = 42$$.
3. *Add one*:
>>$$42+1=43$$.