<s>[INST] The remainder of $1^2+3^2+5^2+7^2+\cdots+1013^2$ divided by $8$ How to find the remainder of $1^2+3^2+5^2+7^2+\cdots+1013^2$ divided by $8$
 [/INST] Hint $\rm\ mod\ 8\!:\,\ odd^2 = \{\pm 1,\, \pm3\}^2 \equiv \color{#C00}1,\:$ so $\rm\,1^2\!+3^2\!+\,\cdots + (2n\!-\!1)^2\equiv \color{#C00}1+\,\cdots\, + \color{#C00}1\equiv\, n\, $
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