Thirteen and variables

Number Theory Level 3

If 13 ∣ 2 x + 3 y 13 \space | \space 2x + 3y , is it true that 13 ∣ 3 x − 2 y 13 \space | \space 3x - 2y ?

Details: ( x , y ) (x,y) are all real numbers.

Notation: a ∣ b a \space | \space b means a a divides b b , or b b is divisible by a a .

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False True

Fidel Simanjuntak by Fidel Simanjuntak , 19, Indonesia

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1 solution

If 13 ∣ 2 x + 3 y 13 \space | \space 2x + 3y , then 13 ∣ 8 ( 2 x + 3 y ) 13 \space | \space 8(2x+3y) . Note that, 13 ∣ 13 x + 26 y 13 \space | \space 13x + 26y .

Then, 13 ∣ 8 ( 2 x + 3 y ) − ( 13 x + 26 y ) 13 \space | \space 8(2x + 3y) - ( 13x + 26y )

⇒ 13 ∣ 3 x − 2 y \Rightarrow 13 \space | \space 3x - 2y .

Hence, it's True \boxed{\text{True}}

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