number properties

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Amrabdelnaby: Master | Next Rank: 500 Posts; Posts: 137; Joined: Fri Nov 13, 2015 11:01 amFollowed by: 2 members; Thanked: 1 times

by Amrabdelnaby » Sun Dec 27, 2015 4:36 am

I need an explanation for for this please.

When a bigger number is subtracted from a smaller number, how can we calculate the units digit?

The last digit of 12^12 + 13^13 - (14^14 Ã— 15^15) =

0
1
5
8
9

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Source: Beat The GMAT — Problem Solving | Sun Dec 27, 2015 4:36 am

chetan.sharma: Senior | Next Rank: 100 Posts; Posts: 46; Joined: Sun Dec 27, 2015 7:03 amFollowed by: 1 member; Thanked: 8 times

by chetan.sharma » Sun Dec 27, 2015 7:15 am

Hi,
Each number follows a certain pattern when it is raised to some power..
It is generally set of 4 unit digits in most cases, and in other less than 4..
so keeping this we can find the last digit ..

now to your Q where you have to subtract a bigger number from a smaller number..
find the numeric value without sign, that is subtract the numeric values..
lets see the Q
The last digit of 12^12 + 13^13 - (14^14 X 15^15) =

0
1
5
8
9

now (14^14 X 15^15) has to have a 0 as last digit..
12^12 will have 6 as last digit as any number with 2 as unit digit will follow 2,4,8,6,2,4,8,...
13^13 will have 6 as last digit as any number with 3 as unit digit will follow 3,9,7,1,3,9,7,1.....
so we have 6+3 being subtracted from 0, which is 1 ans B

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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

by Brent@GMATPrepNow » Sun Dec 27, 2015 4:59 pm

This is a time-consuming question.
We teach how to find units digits of large powers here - https://www.gmatprepnow.com/module/gmat- ... video/1031
Here's a question to practice with - https://www.gmatprepnow.com/module/gmat- ... video/1032

THEN try answering the question you posted.

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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Matt@VeritasPrep: GMAT Instructor; Posts: 2630; Joined: Wed Sep 12, 2012 3:32 pmFollowed by: 119 members; Location: East Bay all the way; Thanked: 625 times; GMAT Score:780

by Matt@VeritasPrep » Sun Dec 27, 2015 5:36 pm

Let's start with 12Â¹Â² + 13Â¹Â³. This will have the same units digit as 2Â¹Â² + 3Â¹Â³, so we can solve that instead.

Notice that the powers of 2 end 2, 4, 8, 6, 2, 4, 8, 6, ...

and that the powers of 3 end 3, 9, 7, 1, 3, 9, 7, 1, ...

So 2Â¹Â² + 3Â¹Â³ ends in 6 + 3, or 9.

Now we're subtracting (14^14 * 15^15). This is a really big number that ends in 0. Notice that 9 - 10 = -1, and that 9 - 20 = -11, and that 9 - 30 = -21, etc.

So something that ends in 9 minus something GREATER that ends in 0 will give us a negative that ends in 1. Success!

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