The tutor investigates a problem involving the remainder of a power.

On page 48 of his Elementary Number Theory, second edition, Underwood Dudley requests the remainder when 2001²⁰⁰¹ is divided by 26.

Solution:

2001 mod 26 = 25 ⇒ 2001 = 26k – 1 for some integer k.

Therefore,

2001²⁰⁰¹ = (26k-1)²⁰⁰¹=(26k-1)(26k-1)(26k-1)…(26k-1)

where there are 2001 brackets of (26k-1) forming the product.

Expanding that product, every term is of the form (26k)^m(-1)^2001-m. Of those, the only term not divisible by 26 will be (26k)⁰(-1)²⁰⁰¹ = -1. Therefore,

2001²⁰⁰¹ = 26q – 1 for some integer q.

⇒2001²⁰⁰¹ mod 26 = -1.

Now, -1 ≡ 25 mod 26 (because adding 26 to -1, you get 25).

So, 2001²⁰⁰¹ mod 26 = 25. Indeed, the remainder when 2001²⁰⁰¹ is divided by 26 is 25.

Source:

Dudley, Underwood. Elementary Number Theory, second ed. New York: W H Freeman and Company, 1978.

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