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The unit's digit of 7 in positive integer power repeats in blocks of 4: {7-9-3-1}. Since 75=4*18+3 then the unit's digit of \(7^{75}\) is the same as the unit's digit of \(7^3\), which is 3.

Therefore the unit's digit of \(7^{75} + 6\) will be: 3 plus 6 = 9.

I think there is an easier/faster method for this one...... we have series of 7,9,3,1,7,9,3,1...... sets of 4 numbers (7,9,3,1)..... closest number to 75 which is divisible by 4 is 76.... therefore 7^76 will have 1 at unit place.........

.that means it will be 3 at the 75th power....so 3+6 will give '9' at unit place for the final answer

for example if we want to find out unit place for 8^9+6

8^1 = 8....8^2=64....8^3=(unitplace)2........8^4=(unitplace)6.....8^5=(unitplace)8....so on......again we have (8,4,2,6,8,4,2,6....) sets of 4.......closest number to 9 which is divisible by 4 is 8.....therefore 8^8 will have 6 at unit place....so 8^9 should have 8 ....and 8+6 gives '4' at unit place for final answer.

the exponents of 7 have units digits ending in 7,9,3 and 1 and then series repeat . So if 7 exponent 75 ( ie 75/4 = 3) which means we need to 3 units digits and add 6 to it Correct answer E

the unit digits in power repeat after an interval of 4. hence, 3^1 has same nits digit as 3^5, 3^9, 3^13 ...... this is true for all integers. hence. 7^75 willhave sam eunits digit as 7^3i.e 3 hence units digit will be 3+6 = 9 option E.