Question

1. Cho C = 5^1 +5^2 + 5^3 +...+ 5^100

Chứng minh rằng :

a. C chia hết cho 31

b. C chia hết cho 39

mọi người giúp mk với ạ^^

Answer 1

a)\(C=5^1+5^2+5^3+....+5^{100}\)

\(\Leftrightarrow C=\left(5^1+5^2+5^3\right)+....+\left(5^{98}+5^{99}+5^{100}\right)\)

\(\Leftrightarrow C=5\left(1+5+5^2\right)+....+5^{98}\left(1+5+5^2\right)\)

\(\Leftrightarrow C=5.31+....+5^{98}.31\)

\(\Leftrightarrow C=31\left(5+5^4+....+5^{98}\right)⋮31\left(dpcm\right)\)

Answer 2

b) \(C=5^1+5^2+5^3+.......+5^{100}\)

\(\Leftrightarrow C=\left(5^1+5^2+5^3+5^4\right)+.......+\left(5^{97}+5^{98}+5^{99}+5^{100}\right)\)

\(\Leftrightarrow C=5\left(1+5+5^2+5^3\right)+.......+5^{97}\left(1+5+5^2+5^3\right)\)

\(\Leftrightarrow C=5.156+.......+5^{97}.156\)

\(\Leftrightarrow C=156\left(5+5^5+5^9+.......+5^{93}+5^{97}\right)\)

\(\Leftrightarrow C=39.4\left(5+5^5+5^9+.......+5^{93}+5^{97}\right)⋮39\left(dpcm\right)\)