\(=5^{20}+\left(5^2\right)^{11}+\left(5^{ }^3\right)^7\)

=\(5^{^{ }20}+5^{22}+5^{21}\)

\(=5^{20}\cdot\left(1+5^2+5^1\right)\)

=\(5^{20}\cdot\left(1+25+5\right)\)

=\(5^{20}\cdot31\)

Vì 31 chia hết chó 31 nên

\(5^{20}+25^{^{ }11}+125^7\)chia hết cho 31

\(^{5^{20}+25^{11}+125^7}\)=\(1.5^{20}+25.25^{10}+\left(5^3\right)^7\)=\(1.5^{20}+25.\left(5^2\right)^{10}+5^{21}\)=\(1.5^{20}+25.5^{20}+5.5^{20}\)

=\(^{5^{20}.\left(1+25+5\right)}\)=\(5^{20}.31\)chia hết cho 31

Vậy \(5^{20}+25^{11}+125^7\)chia hết cho 31

5^20+25^11+125^7=5^20+(5^2)^11+(5^3)^7= 5^20+5^22+5^21=5^20(1+5^2+5)=5^20.31

Vậy 5^20+25^11+125^7 chia hết cho 31

25²⁵ - 25²⁴ = 25²⁴.(25 - 1) = 25²⁴.24 chia hết cho 24

25²⁵ - 25²⁴ = 25²⁴.(25 - 1) = 25²⁴.24 chia hết cho 24

25²⁵ - 25²⁴ = 25²⁴.(25 - 1) = 25²⁴.24 chia hết cho 24

5^25 lẻ ; 2^24 chẵn => 25^25 - 2^24 lẻ => không chia hết cho 24. Đề sai

\(25^{25}-25^{24}=25^{24}.25-25^{24}.1=25^{24}.\left(25-1\right)=25^{24}.24\)chia hết cho 24(đpcm)

\(25^{25}-25^{24}=25^{24}.25-25^{24}=25^{24}.\left(25-1\right)=25^{24}.24\) chia hết cho 24.

\(S=\left(1+2\right)+...+2^6\left(1+2\right)=3\left(1+...+2^6\right)⋮3\)