Divisible by 3?

A number is divisible by 3 if the sum of the digits is divisible by 3. How about a number of the type (22n – 1), where n is a positive integer? Can we show that it is always divisible by 3?
Answer

We can use the following factoring rule:
an – 1 = (a – 1)(an-1 + an-2 + … + a + 1).

We can rewrite 22n – 1 as:

(22)n – 1 = (22 – 1)[ (22)n-1 + (22)n-2 + … + 22 + 1]

= 3(4n-1 + 4n-2 + … + 4 + 1), which is divisible by 3!

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