Prove that the value of the expression (125^2+25^2)(5^2–1) is divisible by 3
2 answers:
Step-by-step explanation:
![A =(125^{2} + 25^{2} ) (5^{2} - 1)\\A = [(5^{3})^{2} + (5^{2})^{2} ] . (5^{2} - 1)\\A = (5^{6} + 5^{4} ). (5^{2} - 1)\\A = [5^{4} . ( 5^{2} + 5)].(5^{2} - 1) \\A = 5^{4} . (25+ 5). (5^{2} - 1)\\A = 5^{4} . 30. (5^{2} - 1)\\](https://tex.z-dn.net/?f=A%20%3D%28125%5E%7B2%7D%20%2B%2025%5E%7B2%7D%20%29%20%285%5E%7B2%7D%20-%201%29%5C%5CA%20%3D%20%5B%285%5E%7B3%7D%29%5E%7B2%7D%20%20%2B%20%285%5E%7B2%7D%29%5E%7B2%7D%20%5D%20.%20%285%5E%7B2%7D%20-%201%29%5C%5CA%20%3D%20%285%5E%7B6%7D%20%20%2B%205%5E%7B4%7D%20%29.%20%285%5E%7B2%7D%20-%201%29%5C%5CA%20%3D%20%5B5%5E%7B4%7D%20.%20%28%205%5E%7B2%7D%20%20%2B%205%29%5D.%285%5E%7B2%7D%20-%201%29%20%5C%5CA%20%3D%205%5E%7B4%7D%20.%20%2825%2B%205%29.%20%285%5E%7B2%7D%20-%201%29%5C%5CA%20%3D%205%5E%7B4%7D%20.%2030.%20%285%5E%7B2%7D%20-%201%29%5C%5C)
Since 30 is divisible by 3
Thus, A is divisible by 3
Good luck!

125 and 25 are integer ⇒125² and 25² are integer ⇒(125²+25²) is integer
One of factors of given expression is 3 so given expression is divisible by 3
q.e.d.
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