Question

Each of $27$ bricks (right rectangular prisms) has dimensions $a \times b \times c,$ where $a, b,$ and $c$ are pairwise relatively prime positive integers. These bricks are arranged to form a $3 \times 3 \times 3$ block, as shown on the left below. A $28$ th brick with the same dimensions is introduced, and these bricks are reconfigured into a $2 \times 2 \times 7$ block, shown on the right. The new block is $1$ unit taller, $1$ unit wider, and $1$ unit deeper than the old one. What is $a + b + c?$

$88$

$89$

$90$

$91$

$92$

Answer: E

Concepts:Diophantine Equation rectangular prism casework

Difficulty rating: 2470

Solution:

Relabel the brick dimensions so the new block has sides $7a,2b,2c.$ These must be the three old side lengths $3a,3b,3c,$ each increased by $1.$ A new side cannot match the old side with the same letter: $7a=3a+1$ has no positive integer solution, while $2b=3b+1$ and $2c=3c+1$ would give negative lengths. Therefore the matching must be one of the two three-cycles. In one orientation, $\begin{aligned} 7a&=3c+1,\\ 2b&=3a+1,\\ 2c&=3b+1. \end{aligned}$ The last two equations give $b=(3a+1)/2$ and $c=(9a+5)/4.$ Substituting into the first gives $28a=27a+19,$ so $(a,b,c)=(19,29,44).$ The other cycle merely exchanges $b$ and $c.$ These lengths are pairwise relatively prime, and $a+b+c=19+29+44=92.$ Thus, E is the correct answer.

2024 AMC 10B Problem 25

Answer: E

Difficulty rating: 2470

Solution:

Problem 25 in Other Years