Question

Chứng minh rằng \(5a^2+15ab-b^2⋮49\Leftrightarrow3a+b⋮7\)

Answer 1

*Nếu a\(⋮\)49 hoặc b\(⋮\)49 => dpcm (*)

* Ta xét Nếu a\(⋮̸\)49 hoặc b\(⋮̸\)49

+ Nếu \(3a+b⋮7\Rightarrow\left(3a+b\right)^2⋮49.\Leftrightarrow A=9a^2+6ab+b^2⋮49\)

B=\(5a^2+15ab-b^2\)

A + B =14a² +21ab = 7a(2a+3b) = 7a(9a+3b-7a) =7.3(3a+b) - 49a².\(⋮\)49 vì 3a+b \(⋮\)7.

A\(⋮\)49 và A+B\(⋮\)49 => B=\(5a^2+15ab-b^2\)\(⋮\)49 (1)

+Nếu B= \(5a^2+15ab-b^2\)\(⋮\)49 => 45a² +15ab+(9a²-b²)-49a²\(⋮\)49

=> 15a(3a+b)+(3a+b)(3a-b)-49a²\(⋮\)49

=>(3a+b)18a-49a² \(⋮\)49 => 3a+b\(⋮\)49 hay 3a+b \(⋮\)7 (2)

(*)(1)(2) => dpcm.