using binomial theorem calculate the remainder when (3)^2010 + (7)^2010 + 2011 is divided by 29

Asked by Yashank Chichra | 16th May, 2013, 11:22: AM

Expert Answer:

By using binomial theorem, we can prove the following result which will be useful in this given question.
Result: an + bn is divisible by a + b only when n is odd.
 
Now, consider 32010 + 72010 
= 32(105) + 72(105) 
= 9105 + 49105 
Now, 105 is odd. So, this expression is divisible by 9+49= 58. Also, 58 is exactly divisible by 29.
Now, consider 2011. It leaves a remainder 10 on division by 29.
Hence,  the remainder when (3)^2010 + (7)^2010 + 2011 is divided by 29 is 10.

Answered by  | 23rd Jun, 2013, 12:56: AM

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