Question

Prove that the value of the expression: (5^18–25^8)(16^4–2^13–4^5) is divisible by 30 and 44.

Answer 1

See explanation

Explanation:

Given the expression

`(5^(18)-25^8)(16^4-2^(13)-4^5)`

First, simplify all terms:

`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)`

Then the expression is

`(5^(18)-5^(16))(2^(16)-2^(13)-2^(10))`

Use distributive property in both factors:

`(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`

Therefore, this expression is divisible by 30 and by 44