Consider the following 100 sets of 10 elements each: \1,2,3,…,10\, \11,12,13,…,20\, \21,22,23,…,30\, \vdots \991,992,993,…,1000\. How many of these sets contain exactly two multiples of 7?

2022 AMC 12B Problem 6 — Solution

2022 AMC 12B Problem 6

Below is the professionally curated solution for Problem 6 of the 2022 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2022 AMC 12B solutions, or check the answer key.

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Concepts:multiple counting integers in a range

Difficulty rating: 1350

Consider the following $100$ sets of $10$ elements each:

$\{1,2,3,\ldots,10\},$ $\{11,12,13,\ldots,20\},$ $\{21,22,23,\ldots,30\},$ $\vdots$ $\{991,992,993,\ldots,1000\}.$

How many of these sets contain exactly two multiples of $7?$

$40$

$42$

$43$

$49$

$50$

Solution:

Among $1$ to $1000$ there are $\left\lfloor \tfrac{1000}{7} \right\rfloor = 142$ multiples of $7.$ Because $10 \gt 7,$ each block of $10$ consecutive integers contains one or two multiples of $7.$

If $x$ blocks contain two and the remaining $100 - x$ contain one, then $2x + (100 - x) = 142,$ so $x = 42.$

Thus, the correct answer is B.

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