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If the number is 17^(256) , find the l...

If the number is ` 17^(256)` , find the last digit

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Since , `17^(256) = (17^(2))^(128) = (289)^(128) = (290-1) ^(128)`
` therefore 17^(156) = ""^(128)C_(0)(290)^(128) - ""^(128)C_(1) (290)^(127) + ""^(128)C_(2) (290)^(126)`
`- ""^(128)C_(3) (290)^(125) +...- ""^(128)C_(125) (290)^(3) + ""^(128)C_(126) (290)^(2) - ""^(128)C_(127) (290) + 1 `
Foe last three digits ,
`17^(256) = (290)^(3) [ ""^(128)C_(0) (290)^(125) - ""^(128)C_(1) (290)^(124) + ""^128C_(2) (290)^(123) - ... - ""^(128)C_(125) (1)] + ""^(128)C_(126) (290)^(2) -""^(128)C_(127) (290) + 1`
` = 1000m + ""^(128)C_(126) (290)^(2) - ""^(128)C_(127) (290) + 1`
where , m is an integer
`= 1000 m + ""^(128)C_(2) (290)^(2) - ""^(125)C_(1) (290)+ 1`
` 1000m + ((128)(127))/(2) (290)^(2) - 128 xx 290 + 1`
` = 1000 m + (128) (127)(290)(145)- (128) (290) + 1`
` = 1000 m + (128) (290) (127xx 145- 1 ) + 1`
` = 1000m + (128) (290) (18414)+ 1`
` 1000m + 683527000 + 680 + 1`
` = 1000 (m + 683527) + 681 `
` therefore ` Last three digits = ` 000 + 681 = 681 `
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ARIHANT MATHS-BIONOMIAL THEOREM-Exercise (Questions Asked In Previous 13 Years Exam)
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  9. The reamainder left out when 8^(2n) - (62)^(2n+1) is divided by 9 is

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  10. For r = 0, 1,"…..",10, let A(r),B(r), and C(r) denote, respectively, t...

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  11. Let S(1) = sum(j=1)^(10) j(j-1).""^(10)C(j), S(2) = sum(j=1)^(10)j."...

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  12. Find the coefficient of x^7 in the expansion of (1 - x -x^2 + x^3)^(6)...

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  13. If n is a positive integer, then (sqrt(3)+1)^(2n)-(sqrt(3)-1)^(2n) is

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  14. The term independent of x in expansion of ((x+1)/(x^(2/3)-x^(1/3)+1)-(...

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  15. The coefficients of three consecutive terms of (1+x)^(n+5) are in the ...

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  16. If the coefficient of x^(3) and x^(4) in the expansion of (1+ax+bx^(2)...

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  17. Coefficient of x^(11) in the expansion of (1+x^2)^4(1+x^3)^7(1+x^4)^(1...

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  18. The sum of coefficient of integral powers of x in the binomial expansi...

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  19. The coefficient of x^9 in the expansion of (1+x)(16 x^2)(1+x^3)(1+x^(1...

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  20. If the number of terms in the expansion of (1-2/x+4/(x^(2)))^n x ne 0,...

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