Each of 27 bricks (right rectangular prisms) has dimensions a×b×c, where a, b, and c are pairwise relatively prime positive integers. These bricks are arranged to form a 3×3×3 block, as shown on the left below. A 28th brick with the same dimensions is introduced, and these bricks are reconfigured into a 2×2×7 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?
By Hints 1, 2, and 3, 2c=3b+1. Substituting the expressions for b and c, 2(37a−1)=3(23a+1)+1. Multiplying both sides by 6, 28a−4=27a+9+6. Therefore, a=19. Now, c=37a−1=37(19)−1=3132=44. Similarly, b=23a+1=23(19)+1=258=29. Thus, Ans=a+b+c=19+29+44=92.
