Find the remainder when 7 to the power 162 is divided by 800?

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Niraj Mishra

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We will apply Remainder Theorem here. Remainder Theorem: Remainder of /P is x Remainder]/P e.g. Remainder of (34 x 36)/32 can be calculated as: Remainder -> 2 Remainder -> 4 so Remainder /32 -> Remainder /32 -> 8 Applying this theorem the current problem: (7^162)/800 can be visualized as...
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We will apply Remainder Theorem here. Remainder Theorem: Remainder of [M x N ]/P is [Remainder[M/P] x Remainder[N/P]]/P e.g. Remainder of (34 x 36)/32 can be calculated as: Remainder[34/32] -> 2 Remainder[36/32] -> 4 so Remainder [34 x 36]/32 -> Remainder [2 x 4]/32 -> 8 Applying this theorem the current problem: (7^162)/800 can be visualized as 7 x 7 x 7 x 7........162 times / 800 now, Remainder [7/800] -> 7 Remainder [7^2/800] -> 49 Remainder [7^3/800] -> 343 Remainder [7^4/800] -> Remainder [2401/800] -> 1 Now, when you see a Remainder of 1, your can now find a pattern. Remainders of the next powers of 7 will form a cycle of 7, 49, 343, 1, 7, 49. 343, 1........ This happens because, as per the Remainder Theorem, 7^5 can be written as 7^4 x 7 so Remainder[7^5/800] -> Remainder [(7^4 x 7)/800] ->Remainder[(1 x 7)/800] -> 7 And so on for 7^6, 7^7, 7^8........ Every time 7 has a power that is a multiple of 4, the remainder will be 1. 162 can be seen as (4*40) + 2 so Remainder [7^162/800] -> Remainder [(7^160) x (7^2)/800] -> Remainder [Remainder[(7^160)/800] x [Remainder[(7^2)/800]]/800-> Remainder [1 x 49]/800 -> So, Remainder will be 49 read less
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