Mathematics: Prove the divisibility of the following numbers: 1. 25^7+5^13 by 30 2. 7^6+7^5−7^4 by 11 3. 5^5−5^4+5^3 by 7

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Prove the divisibility of the following numbers:

dmoney34

17.01.2022

1. Make use of prime factorizations:

$25^7+5^{13}=(5^2)^7+5^{13}=5^{14}+5^{13}$

Pull out a common factor of $5^{13}$ :

$5^{14}+5^{13}=5^{13}(5+1)=5^{13}\cdot6$

Since $30=5\cdot 6$ , it follows that $30\mid25^7+5^{13}$ .

2. Pull out a common factor of 7^4 :

$7^6+7^5-7^4=7^4(7^2+7-1)=7^4\cdot55=5\cdot7^4\cdot11$

3. Pull out a common factor of 5^3 :

$5^5-5^4+5^3=5^3(5^2-5+1)=5^3\cdot21=3\cdot5^3\cdot7$

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