Question

What are the last two digits in the number 11 to the power of 111 ?


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Answer 1

The last two digits in the number $11^{111}$ is 11.

Step-by-step explanation:

We have to find the last two digits in the number 11 to the power of 111.

For this firstly, we have to analyze the pattern for 11 to the power of small numbers, that means;

As we know that; $11^{1}=11$

$11^{2}=121$

$11^{3}=1331$

$11^{4}=14641$

$11^{5}=161051$

As we can clearly see that the last two digits of all these power of 11 are resulting in the second last digit to be the power itself and the last digit is always 1.

This means that if we take the number $11^{111}$, then the last two digits of this will be 11 only because the last four digits will be firstly the power number itself and then the number 1 (1111).

Hence, the last two digits in the number 11 to the power of 111 is 11.

Answer 2

Answer:

11

Step-by-step explanation:

11=10+1

11 to the power of 111 is (10+1) to the power of 111.

Use binomial expansion, we get: C(111,111)*1¹¹¹ + C(111,110)*10*1¹¹⁰+C(111,109)*10²*1¹⁰⁹+.... (the rest items are all multiple of 100).

So the items that really matter are just the first two items, i.e., 11