If k is a positive integer, what is the remainder when (k + 2)(k^3 – k) is divided by 6 ? - The Beat The GMAT Forum

If k is a positive integer, what is the remainder when (k + 2)(k^3 – k) is divided by 6 ?

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BTGModeratorVI: Legendary Member; Posts: 1223; Joined: Sat Feb 15, 2020 2:23 pmFollowed by: 1 member

by BTGModeratorVI » Mon Jun 08, 2020 12:03 pm

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If k is a positive integer, what is the remainder when (k + 2)(k^3 – k) is divided by 6 ?

A. 0
B. 1
C. 2
D. 3
E. 4

Answer: A
Source: Official guide

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Source: Beat The GMAT — Problem Solving | Mon Jun 08, 2020 12:03 pm

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

Re: If k is a positive integer, what is the remainder when (k + 2)(k^3 – k) is divided by 6 ?

by Brent@GMATPrepNow » Mon Jun 08, 2020 12:39 pm

BTGModeratorVI wrote: ↑
Mon Jun 08, 2020 12:03 pm
If k is a positive integer, what is the remainder when (k + 2)(k^3 – k) is divided by 6 ?

A. 0
B. 1
C. 2
D. 3
E. 4

Answer: A
Source: Official guide

One approach is to plug in any positive integer for k and see what happens.

Try k = 1
We get: (k + 2)(k³ – k) = (1 + 2)(1³ – 1)
= (3)(0)
= 0
When we divide 0 by 6 we get 0 with remainder 0
Answer: A

Just for "fun," let's test another k-value
Try k = 2
We get: (k + 2)(k³ – k) = (2 + 2)(2³ – 2)
= (4)(6)
= 24
When we divide 24 by 6 we get 4 with remainder 0
Answer: still A

Try k = 3
We get: (k + 2)(k³ – k) = (3 + 2)(3³ – 3)
= (5)(24)
= 120
When we divide 120 by 6 we get 20 with remainder 0
Answer: still A

And so on...

Brent Hanneson - Creator of GMATPrepNow.com

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8153; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

Re: If k is a positive integer, what is the remainder when (k + 2)(k^3 – k) is divided by 6 ?

by Scott@TargetTestPrep » Wed Jun 10, 2020 12:53 pm

BTGModeratorVI wrote: ↑
Mon Jun 08, 2020 12:03 pm
If k is a positive integer, what is the remainder when (k + 2)(k^3 – k) is divided by 6 ?

A. 0
B. 1
C. 2
D. 3
E. 4

Answer: A
Source: Official guide

Solution:

Let’s simplify the given expression:

(k + 2)(k^3 – k) = (k + 2)[k(k^2 - 1)] = (k + 2)(k)(k + 1)(k - 1)

Reordering the factors in the expression, we have:

(k - 1)(k)(k + 1)(k + 2), which is a product of 4 consecutive integers. Since the product of n consecutive integers is always divisible by n!, the product of 4 consecutive integers is always divisible by 4! = 24 and hence also by 6. Thus, the remainder when (k + 2)(k^3 – k) is divided by 6 is 0.

Answer: A

Scott Woodbury-Stewart
Founder and CEO
[email protected]

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