Prove that of a,b,c are natural numbers

In summary, natural numbers are used in counting and ordering, and are also known as positive integers. To prove that a, b, and c are natural numbers, we must show that they are positive integers without any fractional or decimal components. This is important because it allows us to use specific properties and rules in mathematics. Methods such as using the definition of natural numbers and mathematical induction can be used to prove this. In mathematical proofs, it is safe to assume that a, b, and c are natural numbers as long as it is explicitly stated or given in the context.
  • #1
Caldus
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I would greatly appreciate if someone just at least put me in the right direction with this. I have to prove this:

Prove that of a,b,c are natural numbers, gcd(a,c) = 1 and b divides c, then gcd(a,b) = 1.
 
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  • #2
Write down the defintion of gcd, what properties does gcd have, in particular gdc(x,y) and dividing x and y. Consider if d is a divisor of b then is it a divisor of c?
 
  • #3


To prove this statement, we can use the definition of greatest common divisor (gcd). According to the definition, gcd(a,b) is the largest positive integer that divides both a and b without leaving a remainder. Similarly, gcd(a,c) is the largest positive integer that divides both a and c without leaving a remainder.

Since b divides c, we can express c as a multiple of b, meaning c = kb for some positive integer k. Now, we can substitute this value of c into the definition of gcd(a,c), giving us gcd(a,c) = gcd(a,kb).

Next, we can use the property that gcd(a,bc) = gcd(a,gcd(a,b)) to simplify this expression. This means that gcd(a,kb) = gcd(a,gcd(a,b)).

Since we know that gcd(a,c) = 1, we can substitute this value into our expression, giving us gcd(a,kb) = gcd(a,1). By definition, gcd(a,1) = 1, so we can simplify our expression further to gcd(a,kb) = 1.

Finally, we can use the definition of gcd(a,b) to rewrite this expression as gcd(a,b) = gcd(a,kb) = 1. This proves that if a, b, and c are natural numbers, and gcd(a,c) = 1 and b divides c, then gcd(a,b) = 1.
 

1. What are natural numbers?

Natural numbers are the numbers we use in counting and ordering, such as 1, 2, 3, 4, etc. They are also known as positive integers.

2. How can we prove that a, b, and c are natural numbers?

To prove that a, b, and c are natural numbers, we need to show that they are positive integers, meaning they are greater than or equal to 1 and do not have any fractional or decimal components.

3. Why is it important to prove that a, b, and c are natural numbers?

Proving that a, b, and c are natural numbers is important in mathematics because it allows us to use certain properties and rules that only apply to this specific set of numbers. It also helps us identify which mathematical operations can be performed on these numbers.

4. What methods can be used to prove that a, b, and c are natural numbers?

One method is to show that a, b, and c are positive integers by using the definition of natural numbers. Another method is to use mathematical induction, which involves proving that a statement holds for the first natural number and then showing that if it holds for one natural number, it also holds for the next one.

5. Can we assume that a, b, and c are natural numbers in a mathematical proof?

Yes, we can assume that a, b, and c are natural numbers in a mathematical proof as long as it is explicitly stated in the problem or given in the context. However, it is always recommended to explicitly state the assumptions in order to avoid any confusion or mistakes.

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