Let \(A_{t,k}(n)\) denote the number of partition k-tuples of n with t-cores. In this paper, we prove that if \(n, i\ge 0\) and \(r\ge m\ge 1\) , \(\begin{aligned}&A_{5,5^r\cdot i\!+\!1}(5^mn\!+\!5^m-1)\equiv 0\!\!\!\!\pmod {5^m},\; A_{5,5^r\cdot i+5}(5^mn+5^m-5)\equiv 0\!\!\!\!\pmod {5^{m-1}},\\&A_{5,5^r\cdot i+2}(5^mn+5^m-2)\equiv 0\!\!\!\!\pmod {5^m},\; A_{5,5^r\cdot i+6}(5^mn+5^m-6)\equiv 0\!\!\!\!\pmod {5^{m}},\\&A_{5,5^r\cdot i\!+\!3}(5^mn\!+\!5^m-3)\equiv 0\!\!\!\!\pmod {5^{m\!-\!1}},\; A_{5,5^r\cdot i\!+\!7}(5^mn\!+\!5^m\!-\!7)\equiv 0\!\!\!\!\pmod {5^{m}},\\&A_{5,5^r\cdot i\!+\!4}(5^mn\!+\!5^m-4)\equiv 0\!\!\!\!\pmod {5^{m-1}},\; A_{5,5^r\cdot i\!+\!8}(5^mn\!+\!5^m-8)\equiv 0\!\!\!\!\pmod {5^{m\!-\!1}},\\ \end{aligned}\) from which six conjectures of Saikia et al. (Indian J. Pure Appl. Math., 2024), which were later proved by Tang (J. Ram. Math. Soc., 2025) and four congruences of Liang and Tang (Proc. Edinb. Math. Soc., 2025) follow as special cases.
