Find-the-value-of-X-such-that-1-5-9-13-17-2021-X-mod1000-

Question Number 139737 by mathsuji last updated on 30/Apr/21

F i n d t h e v a l u e o f X s u c h t h a t :

1 \cdot 5 \cdot 9 \cdot 13 \cdot 17 \cdot \dots \cdot 2021 \equiv X (m o d 1000)

Answered by mindispower last updated on 01/May/21

A = \underset{k = 0}{\prod^{505}} (1 + 4 k) \equiv X [1000]

1000 = 8 . 5^{3}

k = 2 n \Rightarrow 1 + 4 k \equiv 1 [8]

k = 2 n + 1 \Rightarrow 1 + 4 k \equiv 5 [8]

\underset{k = 0}{\prod^{505}} (1 + 4 k) \equiv \underset{k = 0}{\prod^{252}} (8 k + 1) . \underset{k = 0}{\prod^{252}} (5 + 8 k) \equiv \underset{k = 0}{\prod^{252}} 5 [8]

= 5^{252} [8]

5^{2} = 1 [8] \Rightarrow 5^{252} \equiv 1 [8]

\Rightarrow A \equiv 1 [8]

1 + 4 k = 5 s \Rightarrow 5 s - 4 k = 1

(s, k) = (1, 1) s o l u t i o n

\Rightarrow 5 (s - 1) - 4 (k - 1) = 0 \Rightarrow k = 5 l + 1

\Rightarrow 1 + 4 (5 l + 1) = 5 (4 l + 1) = 1 + 4 k

l = 0, l = 1, l = 2 \Rightarrow 5, 45, 25 \in {1, 5, 9, 13, 1 \bar{7} \dots . .}

\Rightarrow (1.5.9 \dots \dots 2021) \equiv 0 [125]

A \equiv 0 [125]

A \equiv 1 [8]

A = 8 k + 1 = 125 d

125 d - 8 k = 1

(d, k) = (5, 78) s o l u t i o n

k = 125 m + 78

A \equiv 8 (125 m + 78) + 1 = 1000 m + 625 [1000] \equiv X [100]

X = 625

Commented by mathsuji last updated on 02/May/21

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