Question

Find sum of last 3 digits of 7^9999​

Answer 1

Answer:

this is the answer to find units digit

Step-by-step explanation:

and the last two digits are 4 and 3 respectively

Answer 2

Answer:143

Step-by-step explanation:7^1 = 7

7^2 = 49

7^3 = 343

7^4 = ..401 (mentioning only the last three digits from here on)

7^5 = ...807

7^6 = ...649

7^7 = ...543

7^8 = ... 801

7^9 = ... 607

7^10 = ... 249

7^11 = ...743

7^12 = ...201

7^13 = ...407

7^14 = ... 849

7^15 = ... 943

7^16 = ... 601

7^17 = ...207

7^18 = ... ...449

7^19 = ... 143

7^20 = ... 001

7^21 = ...007

7^22 = ...049

7^23 = ...343

7^24 = ...401

7^25 = ...807

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A clear pattern is noticeable. If, for instance, you are interested only in the last two digits, they repeat every fourth term. Thus the last two digits in 7^2, 7^6, 7^10, 7^14, ..... are 4 and 9. Similarly, the last two digits of 7^3, 7^7, 7^11, 7^15, 7^19, ....... are the same (43).

Similar pattern is noticeable for the last three digits as well. The last three digits repeat every 20th term.

Thus, the last three digits of 7^3 are the same as the last three digits of 7^23,

last three digits of 7^4 are the same as the last three digits of 7^24,

last three digits of 7^5 are the same as the last three digits of 7^25, and so on.

The last three digits of 7^9999 will, therefore, be the same as the last three digits of 7^19 (9999 divided by 20 leaves a remainder of 19).

The last three digits of 7^19 are 1, 4 and 3. Thus, 7^9999 will also have 143 as the last three digits.