What is the remainder when 7^{2024} is divided by 13?

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{"type":"doc","content":[{"type":"paragraph","content":[{"marks":[{"type":"bold"}],"text":"Step 1 (Fermat's Little Theorem):","type":"text"},{"type":"text","text":" Since 13 is prime and "},{"attrs":{"latex":"\gcd(7, 13) = 1"},"type":"math_inline"},{"text":", Fermat's Little Theorem states:","type":"text"}]},{"attrs":{"latex":"7^{12} \equiv 1 \pmod{13}"},"type":"math_block"},{"type":"paragraph","content":[{"text":"Step 2 (Exponent Division):","type":"text","marks":[{"type":"bold"}]},{"text":" Divide the exponent 2024 by 12 to find the remainder:","type":"text"}]},{"attrs":{"latex":"2024 = 12 \times 168 + 8"},"type":"math_block"},{"content":[{"text":"Therefore:","type":"text"}],"type":"paragraph"},{"attrs":{"latex":"7^{2024} \equiv (7^{12})^{168} \times 7^8 \equiv 1^{168} \times 7^8 \equiv 7^8 \pmod{13}"},"type":"math_block"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 3 (Evaluate 7^8 mod 13):","type":"text"}],"type":"paragraph"},{"attrs":{"latex":"7^2 = 49 \equiv 10 \equiv -3 \pmod{13}"},"type":"math_block"},{"type":"math_block","attrs":{"latex":"7^4 \equiv (-3)^2 \equiv 9 \equiv -4 \pmod{13}"}},{"attrs":{"latex":"7^8 \equiv (-4)^2 \equiv 16 \equiv 3 \pmod{13}"},"type":"math_block"},{"content":[{"text":"Thus, the remainder when ","type":"text"},{"attrs":{"latex":"7^{2024}"},"type":"math_inline"},{"text":" is divided by 13 is exactly ","type":"text"},{"attrs":{"latex":"3"},"type":"math_inline"},{"text":".","type":"text"}],"type":"paragraph"}]}

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