1-3-5-7-9-2005-mod-1000-

Question Number 203490 by cortano12 last updated on 20/Jan/24

1 \times 3 \times 5 \times 7 \times 9 \times \dots \times 2005 = \dots (\mod 1000)

Answered by AST last updated on 20/Jan/24

x = 1 \times 3 \times 5 \dots \times 2005 \equiv 0 (m o d 125)

x \equiv {(1 \times 3 \times 5 \times 7)}^{250} \times 1 \times 3 \times 5 (m o d 8) \equiv 7 (m o d 8)

x = 125 q \equiv 7 (m o d 8) \Rightarrow 5 q \equiv 15 (m o d 8) \Rightarrow q \equiv 3 (m o d 8)

1 \times 3 \times 5 \dots \times 2005 = 125 (8 k + 3) \equiv 375 (m o d 1000)

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