The coronal heating mechanism for active region core loops is difficult to determine because these loops are often not resolved and cannot be studied individually. Rather, we concentrate on the "inter-moss" areas between loop footpoints. We use observations from the Hinode EUV Imaging Spectrometer and the X-Ray Telescope to calculate the emission measure distributions of eight inter-moss areas in five different active regions. The combined data sets provide both high- and low-temperature constraints and ensure complete coverage in the temperature range appropriate for active regions. For AR 11113, the emission can be modeled with heating events that occur on timescales less than the cooling time. The loops in the core regions appear to be close to equilibrium and are consistent with steady heating. The other regions studied, however, appear to be dominated by nanoflare heating. Our results are consistent with the idea that active region age is an important parameter in determining whether steady or nanoflare heating is primarily responsible for the core emission, that is, older regions are more likely to be dominated by steady heating, while younger regions show more evidence of nanoflares.