If $a, b,$ and $c$ are all integers divisible by $3$ and $a > b > c > 0,$ then which of the following must be

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Gmat_mission: Legendary Member; Posts: 1622; Joined: Thu Mar 01, 2018 7:22 amFollowed by: 2 members

by Gmat_mission » Thu Oct 08, 2020 4:01 am

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If $a, b,$ and $c$ are all integers divisible by $3$ and $a > b > c > 0,$ then which of the following must be divisible by $3?$

I. $a+b+c$

II. $a-b+c$

III. $\dfrac{abc}9$

A) I only

B) III only

C) I and II

D) I and III

E) I, II, and III

Answer: E

Source: Manhattan GMAT

Source: Beat The GMAT — Problem Solving | Thu Oct 08, 2020 4:01 am

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Re: If $a, b,$ and $c$ are all integers divisible by $3$ and $a > b > c > 0,$ then which of the following must b

by Scott@TargetTestPrep » Mon Oct 12, 2020 4:20 am

Gmat_mission wrote: ↑
Thu Oct 08, 2020 4:01 am
If $a, b,$ and $c$ are all integers divisible by $3$ and $a > b > c > 0,$ then which of the following must be divisible by $3?$

I. $a+b+c$

II. $a-b+c$

III. $\dfrac{abc}9$

A) I only

B) III only

C) I and II

D) I and III

E) I, II, and III

Answer: E

Solution:

Since a, b, and c are all divisible by 3, any sum and difference combo of the three integers will be also divisible by 3. So I and II are true. Furthermore, since each of a, b, and c is divisible by 3, their product is divisible by 3^3 = 27. That is, abc is divisible by 27. Since abc is divisible by 27, abc/9 is divisible by 3 (notice that (abc/9)/3 = abc/27 is an integer). So III is also true.

Answer: E

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