Divisibility problem

[Geronimas]

2. It is given that (3a+7b) is divisible by n and (2a+5b) is divisible by n. Prove that both: a is divisible by n; and b is divisible by n.

 

[Deleted member 4993]

2. It is given that (3a+7b) is divisible by n and (2a+5b) is divisible by n. Prove that both: a is divisible by n; and b is divisible by n.

Please show us what you have tried and exactly where you are stuck.

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Please share your work/thoughts about this assignment.

 

[JeffM]

I shall give you a hint on how to do this by brute force. There probably is a much more elegant way to do it, but elegance is not my strong suit.

All letters stand for integers.

[MATH]n \ | \ (3a + 7b) \text { MEANS } \exists \text { integer } q \text { s.t. } n * q = 3a + 7b.[/MATH]
[MATH]n \ | \ (2a + 5b) \text { MEANS } \exists \text { integer } p \text { s.t. } n * p = 2a + 5b.[/MATH]
From those two statements, can you show

[MATH]\therefore n \ | \ (3a + 7b), \text { and } n \ | \ (2a + 5b) \implies[/MATH]
[MATH]n \ | \ w(3a + 7b),\ n \ | \ w(2a + 5b), \text { and } n \ | \ \{u(3a + 7b) \pm v(2a + 5b)\}.[/MATH]
Those tools will allow you to demonstrate what you must demonstrate.

 

[Dr.Peterson]

In other words, if n divides A and B, then n also divides any multiple of A or of B, and any sum or difference of those. So you can find a way to combine the two expressions to get a or b alone.