Jack and Nancy, immersed in the enchanting realm of numbers, encounter a perplexing riddle that unfolds like a mysterious mathematical tapestry.
"Let us decode this intricate puzzle," Nancy muses, "where numbers dance in multiples of 3, concealing elusive patterns within their digits."
"Behold," she explains, "as we probe the essence of the digits, weaving them into an equation that echoes the melody of divisibility."
"The first number, 74A52B1, whispers secrets as its digits unite, revealing a symphony where A + B resonates with a mesmerizing tune, yielding a cryptic residue of 2 in the harmonic mod 3," she narrates.
"Turning to the second number, 326AB4C, we journey deeper into the cryptic world of numbers," Nancy continues. "The melody we seek within its digits reveals an enigma, echoing the tune of divisibility by 3."
"The symphony culminates in the revelation that C resonates as a residue of 1 in the enigmatic mod 3," she unfolds, "ushering in the possibilities for C as 1, 4, or 7."
"In this numerical quest," Nancy concludes, "we uncover the sum of these elusive values, witnessing the culmination of possibilities as 1, 4, and 7, harmonizing to create the enigmatic sum of 12."
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