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

2022 AMC 12B Problem 6

Attempt Problem 6 of the 2022 AMC 12B below, then check your answer against the professionally curated solution 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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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

A block of

10

consecutive integers contains either one or two multiples of

7

Count the total multiples of

7

\{1,\ldots,1000\},

then let

x

blocks have two and

100-x

have one

Among

1

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.

x

blocks contain two and the remaining

100 - x

contain one, then

2x + (100 - x) = 142,

x = 42.

Thus, the correct answer is B.