In a factory there are two identical solid blocks of iron. When the first block is melted and recast into spheres of equal radii 'r', then 14 cc of iron was left, but when the second block was melted and recast into sphere each of equal radii '2r', then 36 cc of iron was left. The volumes of the solid blocks and all the spheres are in integers. What is the volume of each of the larger spheres of radius '2r'?

Let the total volume of one block be V. The volumes of the spheres must be integers, so the volume of one small sphere (4/3 * pi * r^3) multiplied by an integer n equals V - 14, and the volume of one large sphere (4/3 * pi * (2r)^3 = 8 * 4/3 * pi * r^3) multiplied by an integer m equals V - 36. The volume of the larger sphere is 8 times the volume of the smaller sphere, and since r is an integer, 4/3 * pi * r^3 evaluates to an integer multiple of pi. Testing the given option of 176 cm^3 for the larger sphere gives a smaller sphere volume of 22 cm^3, which allows both V - 14 and V - 36 to be valid integer multiples of these respective sphere volumes, confirming 176 cm^3.

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