Core partition definition confusion

Question

I've seen definitions stating that a $t$-core is a partition with no $t$-hook lengths and that a $t$-core is a partition with no hook lengths divisible by $t$. Are these definitions equivalent? If so, do you have a reference to cite?

Answer 1

The Representation Theory of the Symmetric Group, James and Kerber, doi:10.1017/CBO9781107340732, section 2.7.

A $q$-hook is defined at the start of the section as a hook of length $q$. The following paragraph defines a $q$-core as a diagram $[\alpha]$ which does not contain any $q$-hook, and says that the same name will be used for the corresponding partition $\alpha$. Then 10 pages later we have

2.7.40 The $q$-weight of a diagram equals the number of hook lengths divisible by $q$. In particular, $[\alpha]$ is a $q$-core iff no hook length is divisible by $q$.