1984 AMC 12 Problem 21

Attempt Problem 21 of the 1984 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 1984 AMC 12 solutions, or check the answer key.

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21.

The number of triples

(a,b,c)

of positive integers which satisfy the simultaneous equations

\begin{aligned} ab+bc&=44,\\ ac+bc&=23. \end{aligned}

0

1

2

3

4

Factor the second equation as

c(a+b)=23

Use the primality of

23

and positivity to determine

c

and

a+b

Since

c(a+b)=23

and all variables are positive integers,

c=1

and

a+b=23.

Put

b=23-a

ab+b=44.

Then

(23-a)(a+1)=44,

a^2-22a+21=0,

a=1

21.

Each gives a positive

b,

producing exactly two triples.

Therefore, the correct answer is C.

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