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Three Power Remainder (Posted on 2012-04-14)

Find the remainder when 3²⁰¹² is divided by 2012.

*** For an extra challenge, solve this puzzle without using a computer program.

No Solution Yet Submitted by K Sengupta

Rating: 4.0000 (1 votes)

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Solution (another approach)

| Comment 3 of 4 |

This may be the same as Broll's approach(?)

3²⁰¹² = (3⁴)⁵⁰³ = 3⁴ = 81 (mod 503)   since 503 is prime (Fermat).

So        3²⁰¹² = 503n + 81          (1)

Since all even powers of 3 are congruent to 1 (mod 4), it follows from (1)

that       503n = 0 (mod 4) and therefore that n is divisible by 4.

Let n = 4m, then (1) gives:        3²⁰¹² = 503(4m) + 81 = 2012m + 81,

showing that the required remainder is 81.

Edited on April 15, 2012, 10:51 pm

Posted by Harry on 2012-04-15 19:38:18

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