How Does Modular Arithmetic Prove Divisibility by 17 in Number Theory?

[Gravitational]

Let x and y be integers. Prove that 2x + 3y is divisible
by 17 iff 9x + 5y is divisible by 17.
Solution. 17 | (2x + 3y) ⇒ 17 | [13(2x + 3y)], or 17 | (26x + 39y) ⇒
17 | (9x + 5y), and conversely, 17 | (9x + 5y) ⇒ 17 | [4(9x + 5y)], or
17 | (36x + 20y) ⇒ 17 | (2x + 3y)

Could someone please help me understand this solution. I do not understand it at all. What basis do they have for doing such operations? The solution just doesn't make sense

 

[micromass]

Which part don't you understand??

The only two rules they used were

$$n\vert m~\Rightarrow~n\vert mk$$
and
$$n\vert m,~n\vert k~\Rightarrow~n\vert (m+k)$$

 

[Gravitational]

Why do they multiply 2x+3y by 13? and why do they multiply 9x+5y by 4? why not some other numbers?

 

[Hurkyl]

Why do they multiply 2x+3y by 13? and why do they multiply 9x+5y by 4? why not some other numbers?

So they can get the desired answer. If you use other numbers, you'll get other equivalent statements.

How did they know that would give the desired answer? Trial and error would work. Or, you could try and write down an equation that says "If I multiply by n, then the answer I get is the one I want".

Could someone please help me understand this solution. I do not understand it at all. What basis do they have for doing such operations? The solution just doesn't make sense

You're asking the wrong question, it seems. You don't seem to have meant "I don't understand this solution!" -- you seem to have meant "How could I have come up with this solution myself?"

 

[micromass]

Do you know modulo arithmetic?

 

[Gravitational]

Do you know modulo arithmetic?

no, but i don't think modulo arithmetic is necessary in this problem

 

[Gravitational]

I understand why they multiplied by 13, but i don't see the significance in multiplying by 4

 

[micromass]

no, but i don't think modulo arithmetic is necessary in this problem

It's not necessary in understanding the solution. But it's necessary in understanding why they did what they did.

 

[ramsey2879]

I understand why they multiplied by 13, but i don't see the significance in multiplying by 4

If you understand the first part, then you should understand the logic of the second part. Both parts are required to show the "if and only if" condition. Basically, they used multiplication by 4 since 4*9 = 17*2 plus 2 and that is the way to reduce the 9x to 2x mod 17. Because of the iff part, taking care of the x variable also takes care of the y variable.

 

[Gravitational]

2x + 3y is divisible by 17, there is an integer k such that (2x + 3y)/17 = k <=> 2x + 3y = 17k. Multiply both sides by 13
13(2x + 3y) = 13 * 17k
<=>
26x + 39y = 13 * 17k
<=>
9x + 5y + (17x + 34y) = 13 * 17k
<=> (moving over the thing in the parantheses to the right-hand side and factoring out 17
9x + 5y = 13 * 17k - (17x + 34y) = 13 * 17k - 17(x + 2y) = 17(13k - (x + 2y))

Thus 9x + 5y is divisible by 17.

I actually think this proof is better

 

[Gravitational]

2x + 3y is divisible by 17, there is an integer k such that (2x + 3y)/17 = k <=> 2x + 3y = 17k. Multiply both sides by 13
13(2x + 3y) = 13 * 17k
<=>
26x + 39y = 13 * 17k
<=>
9x + 5y + (17x + 34y) = 13 * 17k
<=> (moving over the thing in the parantheses to the right-hand side and factoring out 17
9x + 5y = 13 * 17k - (17x + 34y) = 13 * 17k - 17(x + 2y) = 17(13k - (x + 2y))

Thus 9x + 5y is divisible by 17.

I actually think this proof is better

solution*

 

[ramsey2879]

2x + 3y is divisible by 17, there is an integer k such that (2x + 3y)/17 = k <=> 2x + 3y = 17k. Multiply both sides by 13
13(2x + 3y) = 13 * 17k
<=>
26x + 39y = 13 * 17k
<=>
9x + 5y + (17x + 34y) = 13 * 17k
<=> (moving over the thing in the parantheses to the right-hand side and factoring out 17
9x + 5y = 13 * 17k - (17x + 34y) = 13 * 17k - 17(x + 2y) = 17(13k - (x + 2y))

Thus 9x + 5y is divisible by 17.

I actually think this proof is better

More powerful if you use the Mod operations. In Mod 17, multiples of 17 == 0.

2x + 3y == 0 Mod 17
26x + 39y == 13* 0 Mod 17
9x + x*0 + 5y + y*0 == 13*0 Mod 17
9x + 5y == 0 Mod 17

I other words 9x+ 5y is divisible by 17 if 2x + 3y is divisible by 17