Sequences: Consecutive Integers - Counting Consecutive Multiples within a Range
Variables $$A$$ and $$B$$ are two multiples of 14, and $$Q$$ is the set of consecutive integers between $$A$$ and $$B$$, inclusive. If $$Q$$ contains nine multiples of 14, how many multiples of 7 are in $$Q$$?
__Alternative method__:
Each multiple of 14 is a multiple of 7, so there are 9 multiples of 7 we already know of. Now, in the interval between any two consecutive multiples of 14 there is an additional multiple of 7.
For example, the interval between 14 and 28 holds 21 as an additional multiple of 7 that we need to count. In the next interval (28 to 42 to the next multiple of 14), there's another hidden multiple of 7: the number 35.
Therefore, to find the total number of multiples of 7, we need to take the known 9 multiples of 14 and count the number of intervals between them. Each such interval holds one more multiple of 7. The secret is that the number of intervals is one less than the number of multiples: between 2 multiples there is only one interval, between 3 multiples we count 2 intervals, etc. Thus, between 9 multiples we'll have 8 intervals, and
>$$9+\color{red}{8}=17$$.
Incorrect.
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Correct.
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To find the multiples of 7 between $$A$$ and $$B$$, you must subtract the
relevant extremes, divide by 7, and add 1. In this case, the difference between the extremes is $$B-A$$.
To make things clearer, you can actually __Plug In__ numbers: a good number for $$A$$ would be 14. Count nine multiples of 14 to find $$B$$. Remember that $$A$$ and $$B$$ are also multiples of 14:
>$$A=\color{red}{14},~ 28,~ 42,~ 56,~ 70,~ 84,~ 98,~ 112,~ \color{red}{126}=B$$.
Then use the method above to find the number of multiples of 7 between 14 and 126, inclusive:
1. Subtract the relevant extremes:
>>$$B-A=126-14=112$$
2. Divide by 7:
>>$$\frac{112}{7}=16$$
3. Add 1:
>>$$16+1=17$$
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