Quandaries
and Queries 

I am currently in the process of taking the GMAT to apply to grad school. I am about to take it for the third time and have a question, one that I have seen on the test every time, that I have never been able to answer. How can one determine the last digit of a large exponent by hand. I have seen several examples of how to do this by breaking it up into smaller exponents, but is there a formula or some common things to look for? Thanks for any help. I am guessing this is a level 10 12 question. Landon 

Hi Landon, There is no formula but there is a pattern that will show you how to find the last digit. For example "What is the last digit of 7^{358}?" Suppose that you are going to construct a table with 358 rows and record last digit of the the 358 powers of 7. Start the table, either using either your calculator or doing the arithmetic by hand.
Notice the pattern of the last digits. They are 7,9,3,1,7,9,3,1,7,9,... The last digit repeats in a pattern that is 4 digits long, 7,9,3,1. If you complete the table for 358 rows how many times will this pattern repeat? 358 divided by 4 is 89 with a remainder of 2 so the pattern will repeat 89 times and then there are two more rows. These rows then have 7 and 9 in the second column so the last digit of 7^{358} is 9. Penny 

