Question

If $c+d=11$ and $c$ and $d$ are positive integers, which of the following is a possible value for $5c+8d$?

• A. $55$
• B. $61$
• C. $69$
• D. $83$
• E. $88$

Answer 1

The correct option is B $61$

Because $c$ and $d$ must be positive integers and $c+d=11$, there are only $10$ possible values for $5c+8d$(starting with $c=1$ and $d=10$,then $c=2$ and $d=9$ and so on). In other words, if your starting point is $5c+8d=58$, where $c=10$ and $d=1$, if you reduce $c$ by $1$ and decrease $d$ by $1$, the resulting sum will increase by $3$; this pattern will continue to occur all the way to your largest possible value $85$. Starting with $58$, then keep adding $3$ until you reach a number found in the answers $58+3=61$ and $61$ is one of the answer choices.

Alternatively, you can notice that consecutive values of $5c+8d$ differ by $3$. In other words, every possible value of $5c+8d$ equals a multiple of $3$ plus some constant. by inspection, you see that the values of $5c+8d$ are all one more than a multiple of $3$; for instance, the value $82=81+1$. The only answer choice that equals a multiple of $3$ plus $1$ is $61$: 60+1$