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A positive integer k is called a "lucky" number if the units digit of k^6 is 4. How many lucky numbers smaller than 1000 are there?

 Nov 17, 2022
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k^6 = something that ends in 4.

Notice that something that ends in 4 has to be e^6 where e is even. 6 wont work because 6^n always ends in 6.

2^6 = 64 that works

2^1 = 2

2^2 = 4

2^3 = 8

2^4 = 6

4^6 = 2^12 = 6 so 4 doesnt work

8^6 = 2^18 = 2^[(4*4) + 2] = 4.

So only 2 and 8 work. 12 and 18 also work. 2, 12, 22, 32... 992. 8, 18, 28... 998.

Each of those sets have 100 elements. So there are 200 lucky numbers.

 Nov 17, 2022

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