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CAT 2020 Slot 1 — QA Question 19

FactorsMedium

If a, b and c are positive integers such that ab = 432, bc = 96 and c < 9, then the smallest possible value of a + b + c is

Answer & solution

Correct answer: 46

  • A

    49

  • B

    56

  • 46

  • D

    59

Solution

Medium

Both products share bb, so abbc=ac=43296=92\dfrac{ab}{bc}=\dfrac{a}{c}=\dfrac{432}{96}=\dfrac{9}{2}, giving a=92ca=\tfrac92 c. Thus cc must be even and less than 99, i.e. c∈{2,4,6,8}c\in\{2,4,6,8\}. For each, compute aa and bb and pick the smallest sum.

1

Relate aa and cc. Divide the two equations.

abbc=43296⇒ ac=92⇒ a=92 c(so c is even)\begin{aligned} &\frac{ab}{bc} = \frac{432}{96}\\ &\Rightarrow\ \frac{a}{c} = \frac{9}{2}\\ &\Rightarrow\ a = \tfrac{9}{2}\,c \quad\text{(so }c\text{ is even)} \end{aligned}
2

Test each even c<9c<9. Then b=96cb=\dfrac{96}{c} (must be an integer) and a=92ca=\tfrac92 c.

c=2: a=9, b=48⇒a+b+c=59c=4: a=18, b=24⇒a+b+c=46c=6: a=27, b=16⇒a+b+c=49c=8: a=36, b=12⇒a+b+c=56\begin{aligned} &c=2:\ a=9,\ b=48 \Rightarrow a+b+c = 59\\ &c=4:\ a=18,\ b=24 \Rightarrow a+b+c = 46\\ &c=6:\ a=27,\ b=16 \Rightarrow a+b+c = 49\\ &c=8:\ a=36,\ b=12 \Rightarrow a+b+c = 56 \end{aligned}
3

Pick the minimum.

min⁡{59,46,49,56}=46(at c=4)\begin{aligned} &\min\{59,46,49,56\} = 46 \quad\text{(at }c=4\text{)} \end{aligned}
a+b+c=46— option (c)a+b+c = 46\quad\text{— option (c)}

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CAT 2020 Slot 1 QA Q19 — Factors solution | TheCATExam