Let \(A(q)=:\sum _{n=0}^{\infty }a_n q^n\) and \(B(q)=:\sum _{n=0}^{\infty }b_n q^n\) be two eta quotients. In some previous papers, the present authors considered the problem of when \(\begin{aligned} a_n=0 \Longleftrightarrow b_n=0. \end{aligned}\) In the present paper we consider the “mod m” version of this problem, i.e. for which eta quotients A(q) and B(q) and for which integers \(m>1\) do we have (non-trivially) that \(\begin{aligned} a_n \equiv 0 \pmod m \Longleftrightarrow b_n \equiv 0 \pmod m? \end{aligned}\) (We say “non-trivially” as there are trivial situations where \(a_n \equiv b_n \pmod m\) for all \(n\ge 0\) ). The m for which we found non-trivial (in the sense just mentioned) results were \(m=p^2\) , \(p=2, 3\) and 5. For \(m=4\) and \(m=9\) , we found results which apply to infinite families of eta quotients. One such is the following: Let A(q) be any eta quotient of the form \(\begin{aligned} A(q) = f_1^{3j_1+1}\prod _{3\not \mid i}f_i^{3j_i}\prod _{3|i}f_i^{j_i} =: \sum _{n=0}^{\infty }a_nq^n, B(q) = \frac{f_3}{f_1^3}A(q) =: \sum _{n=0}^{\infty }b_nq^n \end{aligned}\) with \(f_{k}=\prod _{n=1}^{\infty }(1-q^{kn})\) . Then \(\begin{aligned}&a_{3n}- b_{3n}&\equiv 0\pmod 9, 2a_{3n+1}+b_{3n+1}&\equiv 0 \pmod 9, \\&a_{3n+2}+2b_{3n+2}&\equiv 0 \pmod 9. \end{aligned}\) Some of these theorems also had some combinatorial implications, one example being the following: Let \(p_2^{(3)}(n)\) denote the number of bipartitions \((\pi _1, \pi _2)\) of n where \(\pi _1\) is 3-regular. Then \(\begin{aligned} p_2^{(3)}(n)\equiv 0 \pmod 9 \Longleftrightarrow n \textrm{is not a generalized pentagonal number}. \end{aligned}\) In the case of \(m=25\) , we do not have any general theorems that apply to an infinite family of eta quotients, such as the modulo 9 result stated above. Instead we give two tables of results that appear to hold experimentally. Proofs of results stated in these tables appear to need the theory of modular forms and are more complicated. We do prove some individual results, such as the following: Let the sequences \(\{c_n\}\) and \(\{d_n\}\) be defined by \(\begin{aligned} f_1^{10}=:\sum _{n=0}^{\infty }c_nq^n, \hspace{25pt} f_1^{5}f_5=:\sum _{n=0}^{\infty }d_nq^n. \end{aligned}\) Then \(\begin{aligned} c_n \equiv 0 \pmod {25} \Longleftrightarrow d_n \equiv 0 \pmod {25}. \end{aligned}\)
