Isolate c from the given equation. What are the necessary conditions for c to be a positiveinteger?
D
Practice makes perfect
We are told that c and d are different positiveintegers, for which 4c+d=26. We want to find the sum of all possible values of c. Let's start by isolating c from the given equation.
Since the equation has two different variables, there is not much more that we can do with it. However, remember that c and d are different positive integers, which means they are integers greater than 0.
Integers:& ... - 3, - 2, - 1, 0, 1, 2, 3, ...
Positive Integers:& 1,2,3 ...
Note that if c needs to be a positive integer, then 26-d4 needs to be a positive integer. Therefore, the expression in the numerator, 26-d, has to be divisible by the denominator, 4, so 26-d must be a multiple of 4. Since d is positive and we subtract it from 26, all possible multiples are less than or equal to 26 and greater than 0.
Multiple
Multiple =26-d
d
24
24=26-d
2
20
20=26-d
6
16
16=26-d
10
12
12=26-d
14
8
8=26-d
18
4
4=26-d
22
Now, knowing all possible values of d we can substitute them into the expression 26-d4 to find the possible values of c.
d
26-d/4
c=26-d/4
2
26- 2/4
6
6
26- 6/4
5
10
26- 10/4
4
14
26- 14/4
3
18
26- 18/4
2
22
26- 22/4
1
We have obtained all possible values of c. Recall that c and d have to be different positive integers, but in this case c and d are different in every instance. Finally, we can find the sum of all possible values of c.
6+5+4+3+2+1=21
The obtained sum corresponds to option D.
