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!

moneydude242 Jun 8, 2023

#1**+1 **

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.

Guest Jun 8, 2023