1. Make use of prime factorizations:
[tex]25^7+5^{13}=(5^2)^7+5^{13}=5^{14}+5^{13}[/tex]
Pull out a common factor of [tex]5^{13}[/tex]:
[tex]5^{14}+5^{13}=5^{13}(5+1)=5^{13}\cdot6[/tex]
Since [tex]30=5\cdot 6[/tex], it follows that [tex]30\mid25^7+5^{13}[/tex].
2. Pull out a common factor of [tex]7^4[/tex]:
[tex]7^6+7^5-7^4=7^4(7^2+7-1)=7^4\cdot55=5\cdot7^4\cdot11[/tex]
3. Pull out a common factor of [tex]5^3[/tex]:
[tex]5^5-5^4+5^3=5^3(5^2-5+1)=5^3\cdot21=3\cdot5^3\cdot7[/tex]
