Prove that for all x and y in z, x +3y is a multiple of 7 iff 3x +2y is a multiple of 7. might be to calculate 2(3x +2y)+(x +3y) and/or 4(x +3y)+(3x +2y).

Proof :

First, it is important to have in mind that a number is a multiple of iff there exists such that  .

Also, you have to prove a logical equivalence. To this end, it is possible to prove two logical implications.

Step-by-step explanation:

1.) Let x, y be integers such that x + 3y is a multiple of 7. You have to prove that 3x +2y is a multiple of 7.

In effect, by hypothesis there exists k  such that  x + 3y = 7 k .   So, you get  

.

Therefore, 4(x +3y) + (3x +2y) is a multiple of 7. Then,

.

Given that x,y,k are integers, then x + 2y - 4k is an integer  and hence, 3x + 2y is a multiple of 7.

To finish, it remains to prove its reciprocal statement.

2.) Let x, y be integers such that 3x + 2y is a multiple of 7. You have to prove that  x +3y  is a multiple of 7. Reasoining as before ,   there exists q such that 3x + 2y = 7 \cdot  q.  Thus,

  Thus, is a multiple of 7.

On the other hand,  using the hypothesis .

Finally, thanks that are integer numbers, then is a integer number and therefore, is a multiple of 7.

the answer to this question is yes

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step-by-step explanation: