Edges a, b, and c have lengths that are whole numbers. The surface area of each face is written directly on the face. What are some possible lengths for edges a, b, and c? (Hint: Why can edge a not be 4 m long?)
Let's first consider the formula for the surface area of a rectangular prism:
Surface Area = 2ab + 2bc + 2ac
If the surface areas of all three faces are whole numbers, then 2ab, 2bc, and 2ac must all be even. This means that either a, b, or c must be even. Let's say that a is even. Then, the surface area of the face with sides a and b is an even number, which means that b must also be even. Similarly, the surface area of the face with sides a and c is even, so c must be even as well.
Therefore, if any one of the edges is even, all three edges must be even.
Now, let's consider why edge a cannot be 4 m long. If a = 4, then the surface area of the face with sides a and b is simply 4b, which means that b must be a whole number. The surface area of the face with sides a and c is 4c, which means that c must also be a whole number. However, the surface area of the face with sides b and c is 2bc, which means that bc must be a whole number. This is only possible if either b or c is an even number.
However, we already know that if any one of the edges is even, all three edges must be even. Since a is even, this means that b and c must also be even. But this contradicts our assumption that bc is a whole number, since the product of two even numbers is always even.
Therefore, edge a cannot be 4 m long.