Divisibility

Expert replies

[email protected]: Master | Next Rank: 500 Posts; Posts: 429; Joined: Wed Sep 19, 2012 11:38 pmFollowed by: 4 members; Thanked: 6 times

by [email protected] » Tue Apr 29, 2014 8:10 pm

Hi,

I wanna understand if we can solve for divisibility by 3 the below mentioned number:

2^16-1, I was thinking to solve it by identifying the pattern i.e. 2^16 will have a 5 in the end and if we subtract 1 from it it will give us a 5.

There is smthing wrong I guess I am doing, pls help me understand!

Thanks

Like this post

Source: Beat The GMAT — Data Sufficiency | Tue Apr 29, 2014 8:10 pm

GMAT/MBA Expert

[email protected]: Elite Legendary Member; Posts: 10392; Joined: Sun Jun 23, 2013 6:38 pmFollowed by: 511 members; Location: Palo Alto, CA; Thanked: 2867 times; GMAT Score:800

by [email protected] » Tue Apr 29, 2014 9:37 pm

Hi shibsriz,

This calculation has a couple of pattern shortcuts to it. Here's one way to go about figuring out the pattern:

2^1 = 2 (ends in a 2)
2^2 = 4 (ends in a 4)
2^3 = 8 (ends in an 8)
2^4 = 16 (ends in a 6)

2^5 = 32 (ends in a 2)
2^6 = 64 (ends in a 4)
2^7 = 128 (ends in an 8)
2^8 = 256 (ends in a 6)

Notice the pattern? 2, 4, 8, 6......2, 4, 8, 6.....

This pattern will repeat every 4 terms.

2^16 is 16 terms (or four sets of 4 terms)

2^16 will end in a 6, so 2^16 - 1 will end in a 5

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

Like this post

GMAT/MBA Expert

[email protected]: Elite Legendary Member; Posts: 10392; Joined: Sun Jun 23, 2013 6:38 pmFollowed by: 511 members; Location: Palo Alto, CA; Thanked: 2867 times; GMAT Score:800

by [email protected] » Tue Apr 29, 2014 10:14 pm

Hi shibsriz,

This calculation is also based on an algebra pattern: the difference of squares...and some Number Properties....

2^16 - 1 = (2^8 + 1)(2^8 - 1)

(2^8 - 1) = (2^4 + 1)(2^4 -1)

(2^4 - 1) = (2^2 + 1)(2^2 - 1)

So....

2^16 -1 = (2^8 + 1)(2^4 + 1)(2^2 + 1)(2^2 - 1)

= (257)(17)(5)(3)

Since this is the product of 4 odd numbers (and one of them is a 5), the end result will be an odd multiple of 5.

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

Like this post

Post new topic Post Reply

• Page 1 of 1

GMAT Course Reviews

560 reviews

1168 reviews

431 reviews

359 reviews

147 reviews

5/5

5 Star (560 Reviews)

"Target Test Prep is the closest to the official version of the GMAT exam, about 99% accuracy in terms of the quality and quantity of information. The course has excellently created singular sets of focused lessons and tests for every possible topic that one could come across in the official GMAT exam."

"The TTP course maximizes the efficiency of the time you spend studying. It will take time and effort but I could almost guarantee that if you complete the course exactly as it is laid out you will get an amazing score. They also have a very responsive team willing to help with any questions you might have."

"TTP has two things that I think no other test prep company offers: A teaching approach that reinforces understanding and an attitude that will give you the mental preparedness needed to succeed on the test. TTP gives you a deep understanding of the concept you need to know while teaching you how to think."

Write a Review