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Let \(m\) be a positive three-digit number whose fourth power has the same final three digits as \(m \). Find all possible values of \(m\).

 

Thanks! laugh

 Jun 8, 2023
 #1
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Note that the last three digits of a number are just its value modulo 1000. So, we are looking for a three digit number m such that m4≡m(mod1000).

If m≡0(mod4), then m4≡0(mod1000), so m is a possible value.

If m≡1(mod4), then m4≡1(mod1000). This is not possible, since m is a three digit number.

If m≡2(mod4), then m4≡16≡6(mod1000). This is possible, since 6 is a three digit number.

If m≡3(mod4), then m4≡81≡1(mod1000). This is not possible, since m is a three digit number.

Therefore, the only possible values of m are those that are congruent to 0 modulo 4. These are m=0,4,8,…,96. The smallest value of m greater than 100 is m=80. The largest value of m less than 1000 is m=96. Therefore, the possible values of m are 80,84,88,…,96​.

 Jun 8, 2023
 #2
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m ==376, 625

 

1 - 376^4 ==19,987,173,376

19,987,173,376 mod 1000 ==376

 

2 - 625^4==152,587,890,625

152,587,890,625 mod 1000 ==625

 Jun 8, 2023

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