Page 48 - Elementary Algebra Exercise Book I

ELEMENTARY ALGEBRA EXERCISE BOOK I reAl numBers

1.122 Consider a cuboid whose length, width, height are positive integers

m, n, r and m ≤ n ≤ r . We paint red color on the surface of the cuboid completely and

then chop it into cubes with side length 1. If we know that the number of cubes without
red face plus the number of cubes with two red faces minus the number of cubes with one
red face is 1985, find the values of m, n, r .

Solution: We have three cases, separated by the value of m , to discuss.

(1) If m> 2, then the number of cubes without red face is k 0 =(m − 2)(n − 2)(r − 2),
the number of cubes with one red face is

k 1 = 2(m − 2)(n − 2) + 2(m − 2)(r − 2) + 2(n − 2)(r − 2), the number of cubes with
two red faces is k 2 = 4(m − 2) + 4(n − 2) + 4(r − 2). We have
k 0 + k 2 − k 1 = 1985 ⇒ (m − 2)(n − 2)(r − 2) + 4[(m − 2) + (n − 2)+(r −
2)] − 2[(m − 2)(n − 2) +(m − 2)(r − 2) +(n − 2)(r − 2)] = 1985 ⇒ (m − 2)(n −
2)[(r − 2) − 2] − 2(m − 2)[(r − 2) − 2] − 2(n − 2)[(r − 2) − 2] + 4(r − 2) = 1985 ⇒
(m−2)(n−2)[(r −2)−2]−2(m−2)[(r −2)−2]−2(n−2)[(r −2)−2]+4[(r −2)−2] =

1977 ⇒ [(r − 2) − 2][(m − 2)(n − 2) − 2(m − 2) − 2(n − 2) + 4] = 1977 ⇒ [(r − 2) −
2]{(m − 2)[(n − 2) − 2] − 2[(n − 2) − 2]} = 1977 ⇒ (m − 4)(n − 4)(r − 4) = 1977
Because 1977 = 1 × 3 × 659 = 1 × 1 × 1977 = (−1)(−1) · 1977, then

m − 4 =1,n − 4= 3,r − 4 = 659, or m − 4 =1,n − 4= 1,r − 4 = 1977, or
m − 4= −1,n − 4= −1,r − 4 = 1977. Therefore, m =5,n =7,r = 663, or
m =5,n =5,r = 1981 , or m =3,n =3,r = 1981.

(2) If m =1, then n =1 leads to no solution, thus n ≥ 2. In this case, the number of cubes without

red face k 0 =0, the number of cubes with one red face k 1 =0, and the number of cubes with two

red faces k 2 =(n − 2)(r − 2) . We have k 0 + k 2 − k 1 = k 2 = 1985 , thus

(n − 2)(r − 2) = 1985 = 5 × 397 = 1 × 1985, from which we obtain n − 2 =5,r − 2 = 397,

or n − 2 =1,r − 2 = 1985. Therefore, m =1,n =7,r = 399, or m =1,n =3,r = 1987.

(3) If m =2 , then k 0 =0 , k 1 and k 2 are even numbers. In this case, obviously

k 0 + k 2 − k 1 �= 1985.

Download free eBooks at bookboon.com
48