2024 AMC 10A Problem 23

Problem 23 of 25HarderAlgebra Number Theory

Integers

a,

b,

and

c

satisfy

\begin{aligned} ab + c &= 100, \\ bc + a &= 87, \\ ca + b &= 60. \end{aligned}

What is

ab + bc + ca?

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Solution

Subtract the second equation from the first:

(a-c)(b-1)=13.

Put

u=a-c

and

v=b-1.

The four possibilities

(u,v)=(1,13),

(13,1),

(-1,-13),

and

(-13,-1)

give, after substitution into

bc+a=87,

respectively

c=\frac{86}{15},

c=\frac{74}{3},

c=-8,

and

c=100.

Only the third is an integer solution of all three original equations:

(a,b,c)=(-9,-12,-8).

(The last candidate is

(87,0,100),

which fails

ca+b=60.

) Adding the original equations gives

ab+bc+ca

{}+(a+b+c)=247.

Since

a+b+c=-29,

we obtain

ab+bc+ca

=247-(-29)=276.

Thus, D is the correct answer.