Answer:
See explanation
Step-by-step explanation:
Given the expression
[tex](5^{18}-25^8)(16^4-2^{13}-4^5)[/tex]
First, simplify all terms:
[tex]25^8=(5^2)^8=5^{2\cdot 8}=5^{16}\\ \\16^4=(2^4)^4=2^{4\cdot 4}=2^{16}\\ \\4^5=(2^2)^5=2^{2\cdot 5}=2^{10}[/tex]
Then the expression is
[tex](5^{18}-5^{16})(2^{16}-2^{13}-2^{10})[/tex]
Use distributive property in both factors:
[tex](5^{18}-5^{16})(2^{16}-2^{13}-2^{10})=\\ \\=(5^{16}(5^2-1))(2^{10}(2^6-2^3-1))=\\ \\=(5^{16}(25-1))(2^{10}(64-8-1))=\\ \\=5^{16}\cdot 24\cdot 2^{10}\cdot 55=\\ \\=5^{16}\cdot 4\cdot 6\cdot 2^{10}\cdot 5\cdot 11=\\ \\=5^{16}\cdot 2^{10}\cdot (4\cdot 11)\cdot (5\cdot 6)=\\ \\=5^{16}\cdot 2^{10}\cdot 44\cdot 30[/tex]
Therefore, this expression is divisible by 30 and by 44
