Is 3 a Common Divisor for p and q When 3p² = q²?

In summary, if 3p^2 = q^2 and p and q are integers, we know that q is divisible by 3. However, we cannot prove that p is divisible by 3. We can rewrite the equation as p^2 = q^2/3 and conclude that p has the same prime factors as q, but this does not necessarily mean that p is divisible by 3. We must remember that p and q are integers, not just numbers, and therefore our previous logic may not apply.
  • #1
lordy12
36
0
If 3p^2 = q^2 and p and q are integers, how do I prove that 3 is a common divisor for p and q?
My attempt: q^2 is divisible by 3, so q is divisible by 3. I can't prove that p is divisible by 3.
 
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  • #2
Any integer that is not a perfect square has an irrational square root. What can you do with that?
 
  • #3
Werg22 said:
Any integer that is not a perfect square has an irrational square root. What can you do with that?

Personally, I have no idea what you can do with that.
 
  • #4
lordy12 said:
If 3p^2 = q^2 and p and q are integers, how do I prove that 3 is a common divisor for p and q?
My attempt: q^2 is divisible by 3, so q is divisible by 3. I can't prove that p is divisible by 3.

Now, just re-write it as p^2 = q^2/3. What does that tell you about p?

Remember, p is an integer so p^2 is also an integer. And q is divisible by 3, but there is only one 3. :wink:

Finish it off from there.
 
  • #5
Lol, yeah maybe it's not very relevant. Let me repent:

A power has the same prime factors than its root.
 
  • #6
3|q, rewrite q=3q'
 
  • #7
oops...I completely disregarded the fact that p and q are integers.

sorry
 

FAQ: Is 3 a Common Divisor for p and q When 3p² = q²?

What is divisibility?

Divisibility is the ability of one number to be divided evenly by another number without leaving any remainder.

What is the divisibility rule for 2?

The divisibility rule for 2 is that a number is divisible by 2 if its last digit is an even number (0, 2, 4, 6, or 8).

What is the divisibility rule for 3?

The divisibility rule for 3 is that a number is divisible by 3 if the sum of its digits is also divisible by 3.

What is the divisibility rule for 5?

The divisibility rule for 5 is that a number is divisible by 5 if its last digit is either 0 or 5.

What are p's and q's in relation to divisibility?

P's and q's are commonly used as variables in mathematical equations, and in the context of divisibility, they refer to two different numbers whose relationship is being evaluated (e.g. determining if p is divisible by q).

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