1978 AMC 12 Problem 14

Attempt Problem 14 of the 1978 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 1978 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

14.

If an integer

n,

greater than

8,

is a solution of the equation

x^2-ax+b=0

and the representation of

a

in the base

n

numeration system is

18,

then the base

n

representation of

b

18

20

80

81

280

Translate the base-

n

numeral

18

into

n+8

Use the sum and product of the two roots

In ordinary notation,

a=(18)_n=n+8.

Since one root is

n,

the other root must be

8.

Their product is

b=8n,

whose base-

n

representation is

80.

Therefore, the correct answer is C.

Problem 14 in other years