Quandaries
and Queries 

Who is asking: Student Question: If you raise 7 to the 500th power, what is the numeral that will appear in the units digit position? 

Hi, The first step is to experiment a little. Calculate 57^{2} and 57^{3} = 57^{2} 57 WITHOUT USING A CALCULATOR. Look at what you did. In calculating 57^{2} you found 77=49 and that is all you needed to see that the units digit is 9. Now in calculating 57^{3} you found 79=63 and that is all you needed to see that the units digit of 57^{3} is 3. The point is that the digit 5 in 57 has no influence on the units digit of 57^{28}. To get the correct answer we could ask the seemingly simpler question, if 7 is raised to the 28th power, what must be the units digit in the result? Now I suggest that you look at The units digit of 3 to the 58th power and The units digit of 2 to the power of 31921 to see how these problems are solved. Cheers, 

