Question

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DO 11 CLOD04 W 5000 DOLIUL CLIOUTOU DO 10 DOIS DILIDUL Exercise 19. Adapt the proof of Theorem 30 to show that if n = 2 mod 4 then there is no r e such that p2 = n. This shows, for example, that 10 is irrational. Remarl. 6 Ono con monoralizo the above thoorom to show that if n 7 is

Answer 1

1345 hippere such arenists. Then r=al . (0,b=1 . n. - a = no², n = 2mod 4. - a²=n6²mod 4. Care I a, b are both odd: A a = b ² = 1 mod 4. subsihing auhave 1 En mod 4 but again - n = 2mod 4 ie 12 mod 4. won'ch is a contuadichon care I!: A is odd, bis eren, a²= 1 mod 4. 5 = Omod 4. again, we have 1 Eomody a contuadiction. since n = 2 mod ) Care III a is even, bis odd. then a² = 0 mod 4 ! b = I mod 4. a ² En 62 substuring weget 0= 2 mody (sing which is a contradicion. n=2mod Also both a sb cannot be even,sine (0, 6) = 1 Fence feu any case, cue get a contuadiction so nosuch NET enits.