a) \(A=5+5^2+5^3+....+5^{100}\)

\(5A=5^2+5^3+....+5^{100}+5^{101}\)

\(5A-A=\left(5^2+5^3+5^4+......+5^{101}\right)-\left(5+5^2+5^3+....+5^{100}\right)\)

\(4A=5^{101}-5\)

\(A=\dfrac{5^{101}-5}{4}\)

b) \(A=5+5^2+5^3+....+5^{100}\)

\(A=\left(5+5^2\right)+\left(5^3+5^4\right)+....+\left(5^{99}+5^{100}\right)\)

\(A=\left(5+25\right)+5^2.\left(5+25\right)+....+5^{98}.\left(5+25\right)\)

\(A=30+5^2.30+.....+5^{98}.30\)

\(A=30.\left(1+5^2+....+5^{98}\right)⋮30\)

\(\Rightarrow A⋮30\)

a) A=5+52+53+....+5100

5A=52+53+....+5100+5101

5A−A=(52+53+54+......+5101)−(5+52+53+....+5100)

4A=5101−5

A=5101−54

b) A=5+52+53+....+5100

A=(5+52)+(53+54)+....+(599+5100)

A=(5+25)+52.(5+25)+....+598.(5+25)

A=30+52.30+.....+598.30

A=30.(1+52+....+598)⋮30

⇒A⋮30