Discussion

Find the remainder when \(7^{129}\) is divided by 100.

Answer: C

Correct answer: 7

Work out how quickly the powers of 7 repeat modulo 100.

\[7^1 = 7,\quad 7^2 = 49,\quad 7^3 = 343 \equiv 43,\quad 7^4 = 2401 \equiv 1 \pmod{100}\]

The cycle length is 4, so only the exponent modulo 4 matters. Since \(129 = 4\times 32 + 1\),

\[7^{129} = \left(7^{4}\right)^{32}\times 7 \equiv 1^{32}\times 7 = 7 \pmod{100}\]

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