For how many n in {1,2,3,…,100} is the tens digit of n^2 odd?

1994 AMC 12 Problem 15

Attempt Problem 15 of the 1994 AMC 12 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 1994 AMC 12 solutions, or check the answer key.

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15.

For how many

n

\{1,2,3,\ldots,100\}

is the tens digit of

n^2

odd?

10

20

30

40

50

The parity of the tens digit of

n^2

depends only on the units digit of

n

Check the squares of the ten possible units digits

Write

n=10a+b.

Modulo

100,

n^2\equiv20ab+b^2.

The term

20ab

changes the tens digit by an even amount, so only the tens digit of

b^2

matters. Among

b=0,1,\ldots,9,

it is odd only for

b=4

and

b=6.

Each units digit occurs

10

times from

1

through

100,

giving

2\cdot10=20.

Thus the correct answer is B.