SOLUTION: What is the remainder when 4¹⁰⁰ is divided by 10? For how many different positive integer values of d does (dn – 12)² = 0 have integer solutions?

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Question 753997: What is the remainder when 4¹⁰⁰ is divided by 10?
For how many different positive integer values of d does (dn – 12)² = 0 have integer solutions?
Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
What is the remainder when 4100 is divided by 10?
41 = 4, its last (only) digit is 4.
42 = 16, its last digit is 6.
43 = 64, its last digit is 4.

since 4×4 = 16 and 4×6 = 24, the last digits go

4,6,4,6,4,6,...

The odd powers of 4 end in 4 and the even numbered powers of 4
end in 6

The 100th power or 4 will be an even numbered term of that
sequence so it will end in 6.

When a positive integer is divided by 10 the remainder is 
always the last digit.

Answer: 6

For how many different positive integer values of d does (dn – 12)² = 0 have integer solutions?

These 6 values of d, since (dn - 12)² is 0 when and only when dn = 12

 d  n  dn  (dn - 12)  (dn - 12)²
--------------------------------
 1 12  12      0          0
 2  6  12      0          0
 3  4  12      0          0
 4  3  12      0          0
 6  2  12      0          0
12  1  12      0          0

Answer: 6 values

Edwin
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