Both the roots of x^2 -63x+k=0 are prime numbers. The sum of the digits of k is

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C o n s i d e r space t h e space g i v e n space e q u a t i o n colon x squared minus 63 x plus k equals 0 G i v e n space t h a t space t h e space r o o t s space o f space t h e space a b o v e space e q u a t i o n space a r e space p r i m e space n u m b e r s. L e t space alpha space a n d space beta space a r e space t h e space r o o t s space o f space t h e space e q u a t i o n. W e space k n o w space t h a t space s u m space o f space t h e space r o o t s equals fraction numerator negative b over denominator a end fraction equals fraction numerator negative open parentheses negative 63 close parentheses over denominator 1 end fraction equals 63 rightwards double arrow alpha plus beta equals 63 S u m space o f space t w o space p r i m e space n u m b e r s space i s space a l w a y s space e v e n space e x c e p t space w h e n space o n e space o f space t h e space p r i m e space n u m b e r space i s space t w o. 63 space i s space a n space o d d space n u m b e r. space T h u s comma space alpha equals 2 rightwards double arrow beta equals 63 minus 2 equals 61 rightwards double arrow k equals alpha beta equals 2 cross times 61 equals 122 T h e space s u m space o f space d i g i t s space o f space k space i s equals 1 plus 2 plus 2 equals 5

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