1990 AMC 12 Problem 18

Attempt Problem 18 of the 1990 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 1990 AMC 12 solutions, or check the answer key.

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18.

First

a

is chosen at random from the set

\{1,2,3,\ldots,99,100\},

and then

b

is chosen at random from the same set. The probability that the integer

3^a+7^b

has units digit

8

\frac1{16}

\frac18

\frac3{16}

\frac15

\frac14

The units digits of both powers repeat with period

4

List the residue pairs

(a\bmod4,b\bmod4)

that produce a units digit of

8

The units digits of

3^a

for

a\equiv1,2,3,0\pmod4

are

3,9,7,1,

while those of

7^b

are

7,9,3,1.

A sum ending in

8

occurs for the residue pairs

\begin{aligned} (a,b)\equiv{}&(2,2),(3,0),\\ &(0,1)\pmod4. \end{aligned}

Each residue occurs

25

times among

1,\ldots,100,

so the probability is

\frac{3}{16}.

Thus the correct answer is C.

Problem 18 in other years