Question

Using binomial theorem, find the remainder when \(5^{103}\) is divided by 13.

Answer 1

We have, 5¹⁰³ 

= 5.5¹⁰² = 5(26 – 1)⁵¹

 Now expanding the above by binomial theorem we get, 

= [ ⁵¹C₀ (26)⁵¹ 1⁰ – ⁵¹C₁ (26)⁵⁰ 1¹ +…+ ⁵¹C₅₁ (26)⁰ 1⁵¹]

 In the above expansion, all terms except the constant (last) term will contain 26, which is divisible by 13 

Hence the remainder is 

5(– 1⁾⁵¹ = – 5 

Which is the same as remainder being 8.