Question

Write the last two digits of the number 3⁴⁰⁰

Answer 1

Given, 3⁴⁰⁰ 

= (1 + 2)⁴⁰⁰ 

=⁴⁰⁰ C₀ (1)^{400 – 0} (2)⁰ + ⁴⁰⁰ C₁ (1)^{400 – 1} (2)^{1 + 400}C₂ (1)^{400 – 2} (2)² +…+ ⁴⁰⁰ C₄₀₀ (1)^{400 – 400}(2)⁴⁰⁰ 

We observe that all term expect the first one are divisible by 100, so the remainder when 3 ⁴⁰⁰ + 100 is same as the [ ⁴⁰⁰ C₀ (1)⁴⁰⁰ (2)⁴⁰⁰  ] ÷ 100 = 1 ÷ 100