2015 AMC 10B Problem 25

Question

A rectangular box measures

a \times b \times c,

where

a,

b,

and

c

are integers and

1\leq a \leq b \leq c.

The volume and the surface area of the box are numerically equal. How many ordered triples

(a,b,c)

are possible?

4

10

12

21

26

Start from

abc=2(ab+ac+bc)

Fix

a\le6

and factor the remaining equation

The condition is

abc=2(ab+ac+bc).

Since

abc\le6bc

, we have

a\le6

. Also

a=1

and

a=2

give no positive solutions, so test

a=3,4,5,6

.

For

a=3

,

(b-6)(c-6)=36

, giving

(b,c)=(7,42),

(8,24),

(9,18),

(10,15),

(12,12)

. For

a=4

,

(b-4)(c-4)=16

, giving

(5,20),(6,12),(8,8)

.

For

a=5

,

(3b-10)(3c-10)=100

, and the only solution with

a\le b\le c

is

(b,c)=(5,10)

. For

a=6

,

(b-3)(c-3)=9

, and the only solution with

a\le b\le c

is

(6,6)

.

The total number of triples is

5+3+1+1=10

.

Thus, the correct answer is B.

Problem 25 in other years