Question 753997
What is the remainder when 4<sup>100</sup> is divided by 10?
<pre>
4<sup>1</sup> = 4, its last (only) digit is 4.
4<sup>2</sup> = 16, its last digit is 6.
4<sup>3</sup> = 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
</pre>
For how many different positive integer values of d does (dn – 12)² = 0 have integer solutions?
<pre>

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</pre>