1964 AMC 12 Problem 25

Attempt Problem 25 of the 1964 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 1964 AMC 12 solutions, or check the answer key.

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25.

The set of values of

m

for which

x^2+3xy+x+my-m

has two factors, with integer coefficients, which are linear in

x

and

y,

is precisely:

0,

12,

-12

0,

12

12,

-12

12

0

Write the factors as

(x+ay+b)(x+cy+d)

The missing

y^2

-term and the

3xy

-term force

\{a,c\}=\{0,3\}

Write the factorization as

(x+ay+b)(x+cy+d).

Comparing the

y^2

and

xy

coefficients gives

ac=0

and

a+c=3,

so take

a=0,

c=3.

The other coefficients give

\begin{aligned} b+d&=1,\\ bd&=-m,\\ 3b&=m. \end{aligned}

Hence

b(d+3)=0.

b=0,

then

m=0;

otherwise

d=-3,

b=4,

and

m=12.

Both values produce valid integer factorizations.

Therefore, the correct answer is B.

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