there will be a patten to the units digit.
7^0 = 1 last digit = 1
7^1 = 7 last digit = 7
7^2 = 49 last digit = 9
7^3 = 343 last digit = 3
7^4 = 2401 last digit = 1 ***** (pattern starts to repeat itself here)
7^5 = 16807 last digit = 7
7^6 = 117649 last digit = 9
7^7 = 8234543 last digit = 3
7^8 = 5764801 last digit = 1 ***** (pattern repeats again here)
7^9 = 40353607 last digit = 7
7^10 = 282475249 last digit = 9
7^11 = 1977326743 last digit = 3
7^12 = 13841287201 last digit = 1 ***** (pattern repeats again here)
the pattern repeats itself every 4 exponents.
make a rule from this and apply it to the larger exponent.
7^0 = 1
7^4 = 1
7^8 = 1
if you divide the exponent by 4, you will get the first digit in the sequence.
4/4 = 1.0 which means the last digit is 1
5/4 = 1.25 which means the last digit is 7
6/4 = 1.5 which means the last digit is 9
7/4 = 1.75 which means the last digit is 3
8/4 = 2.0 which means the last digit is 1
9/4 = 2.25 which means the last digit is 7
10/4 = 2.5 which means the last digit is 9
11/4 = 2.75 which means the last digit is 3
12/4 = 3.0 which means the last digit is 1
this is substantiated by the table above.
the rule can be developed as follows:
if the fractional part is 0, then the last digit is 1
if the fractional part is .25, then the last digit is 7
if the fractional part is .5, then the last digit is 9
if the fractional part is .75, then the last digit is 3
applying that rule to your problem, we get:
7^155 results in a last digit of:
155/4 = 38.75
the fractional part is .75 so the last digit is 3.