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CBSE Class 10 Answered

Find the HCF of 2^100– 1 and 2^120– 1
Asked by rushabhjain.a | 17 Apr, 2019, 02:30: PM
answered-by-expert Expert Answer
G i v e n space t w o space n u m b e r s space a r e space 2 to the power of 100 minus 1 space a n d space 2 to the power of 120 minus 1 N o w comma space 2 to the power of 100 equals left parenthesis 2 to the power of 20 right parenthesis to the power of 5 space a n d space 1 equals 1 to the power of 5 rightwards double arrow 2 to the power of 100 minus 1 equals open parentheses 2 to the power of 20 close parentheses to the power of 5 minus 1 to the power of 5 W e space k n o w space t h a t comma space open parentheses x to the power of n minus y to the power of n close parentheses space i s space d i v i s i b l e space b y space left parenthesis x minus y right parenthesis space n element of N space.... space open parentheses 1 close parentheses rightwards double arrow left parenthesis 2 to the power of 20 right parenthesis to the power of 5 minus 1 to the power of 5 space i s space d i v i s i b l e space b y space left parenthesis 2 to the power of 20 minus 1 right parenthesis rightwards double arrow 2 to the power of 100 minus 1 space i s space d i v i s i b l e space b y space left parenthesis 2 to the power of 20 minus 1 right parenthesis space... space open parentheses 2 close parentheses A l s o comma space space 2 to the power of 120 equals open parentheses 2 to the power of 20 close parentheses to the power of 6 rightwards double arrow space 2 to the power of 120 minus 1 equals open parentheses 2 to the power of 20 close parentheses to the power of 6 minus 1 to the power of 6 U sin g space left parenthesis 1 right parenthesis comma space w e space h a v e open parentheses 2 to the power of 20 close parentheses to the power of 6 minus 1 to the power of 6 space i s space d i v i s i b l e space b y space left parenthesis 2 to the power of 20 minus 1 right parenthesis rightwards double arrow 2 to the power of 120 minus 1 space i s space d i v i s i b l e space b y space left parenthesis 2 to the power of 20 minus 1 right parenthesis space... space open parentheses 3 close parentheses rightwards double arrow 2 to the power of 100 minus 1 space a n d space 2 to the power of 120 minus 1 space a r e space d i v i s i b l e space b y space left parenthesis 2 to the power of 20 minus 1 right parenthesis A l s o comma space left parenthesis 2 to the power of 20 minus 1 right parenthesis space i s space t h e space g r e a t e s t space n u m b e r space w h i c h space c a n space d i v i d e space b o t h space t h e space g i v e n space n u m b e r s H e n c e comma space left parenthesis 2 to the power of 20 minus 1 right parenthesis space i s space H C F space o f space 2 to the power of 100 minus 1 space a n d space 2 to the power of 120 minus 1.
Answered by Renu Varma | 18 Apr, 2019, 10:04: AM

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