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Minimum value of a sum

Source: 2009 AIME I #14

March 18, 2009
linear algebramatrixmodular arithmeticAMC

Problem Statement

For t \equal{} 1, 2, 3, 4, define \displaystyle S_t \equal{} \sum_{i \equal{} 1}^{350}a_i^t, where ai∈{1,2,3,4} a_i \in \{1,2,3,4\}. If S_1 \equal{} 513 and S_4 \equal{} 4745, find the minimum possible value for S2 S_2.
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