if a number divides one of two numbers but not the other, then it will not divide their sum
This statement is not true. A number can divide one of two numbers but not the other and still divide their sum.
For example, let's consider the numbers 6, 8, and 14.
6 divides 6 and 14 (6/6 = 1 and 6/14 = 0.428571...), but it does not divide 8 (6/8 = 0.75).
However, 6 divides the sum of 6 and 8 (6 + 8 = 14, and 14/6 = 2), contradicting the statement.
To prove that if a number divides one of two numbers but not the other, then it will not divide their sum, we can use a proof by contradiction.
Let's assume that there is a number, let's call it "x", that divides one of two numbers, say "a", but not the other number, say "b". Now, let's suppose that this number "x" also divides the sum of these two numbers, which is "a + b".
By definition, if "x" divides "a", then there exists an integer "k" such that "a = x * k". Similarly, if "x" divides "a + b", then there exists an integer "m" such that "a + b = x * m".
Now, let's subtract the equation "a = x * k" from the equation "a + b = x * m":
(a + b) - a = x * m - x * k
b = x * (m - k)
Since both "m" and "k" are integers, the quantity "(m - k)" is also an integer. Therefore, we have expressed "b" as the product of "x" and an integer "(m - k)".
This shows that "x" divides "b" as well, which contradicts our initial assumption that "x" does not divide "b".
Hence, our assumption that "x" divides the sum of two numbers "a + b" must be false.
Therefore, if a number divides one of two numbers but not the other, then it will not divide their sum.
To determine whether a number divides another number, you can use the concept of divisibility. Let's say we have three numbers - A, B, and C.
If a number, let's call it N, divides one of two numbers, let's say it divides A but not B, we can write it as:
A = N * X,
B = N * Y,
where X and Y are integers.
Now, let's consider their sum, A + B:
A + B = N * X + N * Y
= N * (X + Y)
Since X and Y are integers, X + Y is also an integer. Therefore, we can rewrite the sum as:
A + B = N * (X + Y)
Based on this, we can see that the sum of A and B is a multiple of N. Since N is the common factor of A and B, it will always divide their sum. Thus, it is not correct to say that if a number divides one of two numbers but not the other, then it will not divide their sum.