If m is divisible by n, we can rewrite m as the product of n and some factor.
Is a+b Divisible by c? Yes Is b-a Divisible by c? Yes Explanation: See solution.
Practice makes perfect
We are given that there are two whole numbers a and b, both of which are divisible by c. Let's recall what it means if one number is divisible by the other number.
If m is divisible by n, we can rewrite m as the product of n and some factor.
In our case, this means that we can rewrite a and b as products of c and some other whole numbers m and n. Additionally, because b > a, the value of n must be greater than m.
a &= c* m
b &= c* n
We want to determine whether a+b and b-a are divisible by c. To do this, we can use the equations we just wrote. Let's start with a+b.
Because we are able to rewrite b-a as the product of c and a positive whole number n-m, the difference b-a is also divisible by c.
Extra
Example Using Numbers
We can make sure that our reasoning is correct by using an example. Let's assume that a is equal to 6 and b is equal to 9. Both 6 and 9 are whole numbers and 9 > 6, so our example numbers satisfy the given conditions.
a &= 6
b &= 9
Both of our numbers are divisible by 3, so let's make our value of c 3. This means we can rewrite our numbers as products where one factor is 3.
6 &= 3* 2
9 &= 3 * 3
Now let's analyze whether the sum and the difference of our numbers are also divisible by 3.
Expression
Substitute
Factored Form
6+ 9
3* 2+ 3 * 3
3(2+3)
9- 6
3* 3- 3 * 2
3(3-2)
We were able to rewrite a+b and b-a as the product of 3 and some other factor. This means that for our example a, b, and c, both a+b and b-a are divisible by c.
