<PRE><B>Question 255255</B>
7^1 = 7
7^2 = 49
7^3 = 343
7^4 = 2401
7^5 = 16807
7^6 = 117649
7^7 = 823543
7^8 = 5764801
7^9 = 40353607
7^10 = 282475249
7^11 = 1977326743


the last digit is repeating in a pattern of:


7,9,3,1,
7,9,3,1,
7,9,3,1,


to find out what the last digit is, you need to make a formula.


the formula would be taking the larger exponent and breaking it down into the smaller exponent


example:


suppose your number is 7^6.


6/4 = 2 and 7^2 = 49 so the last digit is 9.


look at 7^6 above and you can see that the last digit is 9.


suppose your number is 7^11.


11/4 = 2 with a remainder of 3.


7^3 = 343 so the last digit is 3.


look up 7^11 above and you can see that the last digit is 3.


you can use this formula for larger numbers even though you can&#39;t check them because the calculator doesn&#39;t carry enough digits.


I used excel and the highest I could go was 7^17 as shown below:


&lt;pre&gt;
base	exponent	number	           last digit
7	1	        7	           7
7	2	       49	           9
7	3	       343	           3
7	4	       2401	           1
7	5	       16807	           7
7	6	       117649	           9
7	7	       823543	           3
7	8	       5764801	           1
7	9	       40353607	           7
7	10	       282475249	   9
7	11	       1977326743	   3
7	12	       13841287201	   1
7	13	       96889010407	   7
7	14	       678223072849	   9
7	15	       4747561509943	   3
7	16	       33232930569601	   1
7	17	       232630513987207	   7
7       18                                 9
7       19                                 3
7       20                                 1 
7       21			           etc.....        
&lt;/pre&gt;


for a final check, try 7^16.


16/4 = 4 with a remainder of 0.


7^0 = 1 so the remainder should be 1.


it is, as shown in the table from excel above.


note that 16 / 4 = 4 with a remainder of 0.   it also equals 3 with a remainder of 4 which confirms that the last digit is 1.


In your problem, the number is 7^2005


divide 2005 by 4 to get 501 with a remainder of 1.


7^1 = 7, so the last digit of 7^2005 = 1

</PRE>