the ans. is 081

 

solution:-->

 

17²⁵⁶ can be re-written as (7+10)²⁵⁶

 

using binomial expansion & writting just the 1st three terms becoz' just the 1st three terms will give the answer --->

 

²⁵⁶C₀ 7²⁵⁶ 10⁰ + ²⁵⁶C₁ 7²⁵⁵ 10¹ + ²⁵⁶C₂ 7²⁵⁴ 10² +.....

 

now writing the last digit of powers of 7

7^4K     => 01,

7^4K+1 => 07,

7^4K+2 => 49,

7^4K+3 => 43,  where 'K' is any whole number.

 

now from expansion,

the 1st term gives us 601 as last 3 digits, so last digits of 17²⁵⁶ is 601

 

the 2nd term gives us 080 as the last digits , so last digits of 17²⁵⁶ are 580 + 601 = 681

 

similarly, from the 3rd term, we get zero as last digit when we solve ²⁵⁶C₂ 7²⁵⁴ & therefore, the last 3 digits will be 000

=> last 3 digits of 17²⁵⁶ will be 681

 

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