Question 615553
142 = 42 (mod 100)


142^2 = 142*142 = 42*42 = 1764 = 64 (mod 100)


144^4 = (144^2)^2 = 64^2 = 4096 = 96 (mod 100)


144^8 = (144^4)^2 = 96^2 = 9216 = 16 (mod 100)


144^16 = (144^8)^2 = 16^2 = 256 = 56 (mod 100)


144^32 = (144^16)^2 = 56^2 = 3136 = 36 (mod 100)


144^64 = (144^32)^2 = 36^2 = 1296 = 96 (mod 100)


144^128 = (144^64)^2 = 96^2 = 9216 = 16 (mod 100) ... Notice that there's a pattern starting to emerge


144^256 = (144^128)^2 = 16^2 = 256 = 56 (mod 100)

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So 


142^2 = 64 (mod 100)


144^4 = 96 (mod 100)


144^8 = 16 (mod 100)


144^16 = 56 (mod 100)


144^32 = 36 (mod 100)


144^64 = 96 (mod 100)


144^128 = 16 (mod 100) 


144^256 = 56 (mod 100)


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142^291 (mod 100)


142^(256 + 32 + 2 + 1) (mod 100)


( 142^256 ) * ( 142^32 ) * ( 142^2 ) * ( 142^1 ) (mod 100)


( 56 ) * ( 36 ) * ( 64 ) * ( 42 ) (mod 100)


5419008 (mod 100)


8 (mod 100)



Therefore 142^291 = 8 (mod 100)



So the last two digits of 142^291 are <font color="red">08</font>