# Help with a proof on divisibility

• AlexChandler
In summary, the conversation discusses how to prove the statement that if 5 divides a^2 + b^2 + c^2, then 5 must also divide a, b, and c. The conversation starts with a possible method of proof and then realizes that the statement is incorrect and should instead prove that 5 divides a or 5 divides b or 5 divides c. The conversation then discusses how to find a counterexample and hints at a possible method for proving the correct statement.
AlexChandler

## Homework Statement

Prove the following:
If 5 divides $$a^2 + b^2 + c^2$$ then 5 divides a and 5 divides b and 5 divides c.

## Homework Equations

$$5 \mid a \implies a=5k , k \in Z$$

## The Attempt at a Solution

My idea is to assume 5 divides $$a^2 + b^2 +c^2$$

also assume that "5 does not divide a" or "5 does not divide b" or "5 does not divide c"

then by the division algorithm, for some integers k,s,t

$$a=5k+1$$ or $$a=5k+2$$ or $$a=5k+3$$ or $$a=5k+4$$

$$b=5s+1$$ or $$b=5s+2$$ or $$b=5s+3$$ or $$b=5s+4$$

$$c=5t+1$$ or $$c=5t+2$$ or $$c=5t+3$$ or $$c=5t+4$$

Now if we square each possibility for a, b, and c and add them together in each possible way, we would find that there is no possible combination that will give that 5 divides a^2 + b^2 + c^2. Then we would see that this is a contradiction so we must have that 5 divides a,b, and c. However, there must be an easier way than to have to manipulate 3^4 (i think) equations. Does anybody see a better way to go about this? or a quicker way to check all of the equations?
Thanks!

Last edited:
Are you certain that your theorem is even correct?? Have you tried finding counterexample?

You are absolutely right! I misread the problem. It says prove that 5 divides a or 5 divides b or 5 divides c. Thanks! This is much easier! Hah i feel stupid

You probably want to try this on your own. But it may be worth to see what kind of remainders a2 can have when divided by 5...

Yes that is exactly right. I have it figured out now. Thank you

## What is divisibility?

Divisibility is the mathematical concept of being able to divide one number by another without any remainder. In other words, a number is divisible by another number if the result is a whole number or integer.

## How can I prove divisibility?

One way to prove divisibility is by using the division algorithm, which states that for any two integers a and b, with b not equal to 0, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < |b|.

## What is the difference between divisibility and factors?

Divisibility is the ability to divide one number by another without any remainder, while factors are numbers that can be multiplied together to get a given number. For example, 2 is a factor of 6, but 6 is not divisible by 2.

## Can I use the divisibility rule for any number?

Yes, there are different divisibility rules for different numbers. For example, the divisibility rule for 2 is that a number is divisible by 2 if its last digit is even. However, not all numbers have a simple divisibility rule.

## How can I use divisibility in real life?

Divisibility is used in various real-life situations, such as dividing a pizza evenly among friends, calculating discounts and taxes, and converting fractions to decimals. It is also used in advanced mathematical concepts and applications, such as cryptography and prime factorization.

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