We establish several partition identities with distinct colors that arise from Ramanujan's theta-function identities and formulas for multipliers in the theory of modular equations. Also, we deduce few partition congruences as a corollary of some partition identities.

Let pod 2 (n) denote the number of 2-regular partitions of n with distinct odd parts (even parts are unrestricted). In this article, we obtain congruences for pod 2 (n) mod 2 and mod 8 using some generating function manipulations and the theory of Hecke eigenform.

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Let $b_{3,5}(n)$ denote the number of partitions of $n$ into parts that are not multiples of 3 or 5. We establish several infinite families of congruences modulo 2 for $b_{3,5}(n)$. In the process, we also prove numerous parity results for broken 7-diamond partitions.

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