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

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

Text Solution

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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 `
For last digit
`17^(256) = 290 [ ""^(128)C_(0) (290)^(127) - ""^(128)C_(1) (290)^(290)^(126)`
` + ""^(128)C_(2) (290)^(125) - ... - (128) C_(127) (1)] + 1 `
` = 290 (k) + 1 ` where k is an integer .
` therefore` Last digit = 0 + 1 =
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