A repdigitb is a number n that, when represented in base b, consists of the same symbol written over and over, repeated at least twice. (See p. 714.) For example, 666 is a repdigit10: when you write...

A repdigitb is a number n that, when represented in base b, consists of the same symbol written over and over, repeated at least twice. (See p. 714.) For example, 666 is a repdigit10: when you write [666]10, it’s the same digit (“6”) repeated (in this case, three times). One way of understanding that 666 is a repdigit10 is that 666 = 6 + 60 + 600 =6 10⁰
+ 6 10¹
+ 6 10²
. We can write [40]10 as [130]5 because 40 = 0 + 3 5 + 1 5 2 , or as [101000]2 because 40 = 1 2 3 + 1 2 5 . So 40 is not a repdigit10, repdigit5, or repdigit2. But 40 is a repdigit3, because 40 = [1111]3.

1.Prove that every number n ≥ 3 is a repdigitb for some base b ≥ 2, where n = [11 1]b

.

2.Prove that every even number n > 6 is a repdigitbfor some base b ≥ 2, where n = [22 2]b

3. Prove that no odd number n is a repdigitb of the form [22 2]b, for any base b