Remainder Problem

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jnellaz: Master | Next Rank: 500 Posts; Posts: 114; Joined: Mon Oct 20, 2008 7:03 pmFollowed by: 5 members; Thanked: 4 times

by jnellaz » Fri Nov 07, 2008 9:46 am

I came across this problem but I am trying to figure out how they got to the answer. Can someone provide their approach to this problem? Thanks!

If n is a positive integer, what is the remainder when 38n+3 + 2 is divided by 5?
A. 0
B. 1
C. 2
D. 3
E. 4

[spoiler]Answer: E or 4[/spoiler][spoiler]
[/spoiler]

Last edited by jnellaz on Fri Nov 07, 2008 10:08 am, edited 2 times in total.

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Source: Beat The GMAT — Problem Solving | Fri Nov 07, 2008 9:46 am

logitech: Legendary Member; Posts: 2134; Joined: Mon Oct 20, 2008 11:26 pmFollowed by: 25 members; Thanked: 237 times; GMAT Score:730

Re: Remainder Problem

by logitech » Fri Nov 07, 2008 9:59 am

jnellaz wrote:I came across this problem but I am trying to figure out how they got to the answer. Can someone provide their approach to this problem? Thanks!

If n is a positive integer, what is the remainder when 38n+3 + 2 is divided by 5?
A. 0
B. 1
C. 2
D. 3
E. 4

[spoiler]Answer: 4[/spoiler][spoiler]
[/spoiler]

Two things:

1) There seems to be a typo
2) Does answer 4 mean 4 or D ?

LGTCH
---------------------
"DON'T LET ANYONE STEAL YOUR DREAM!"

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jnellaz: Master | Next Rank: 500 Posts; Posts: 114; Joined: Mon Oct 20, 2008 7:03 pmFollowed by: 5 members; Thanked: 4 times

by jnellaz » Fri Nov 07, 2008 10:11 am

Hey there,

I fixed the spoiler function to show the correct answer choice - but the answer is 4 or choice E.

Where do feel there is a typo? I got this question from here:
https://www.urch.com/forums/gmat-problem ... -help.html

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logitech: Legendary Member; Posts: 2134; Joined: Mon Oct 20, 2008 11:26 pmFollowed by: 25 members; Thanked: 237 times; GMAT Score:730

by logitech » Fri Nov 07, 2008 10:38 am

jnellaz wrote:Hey there,

I fixed the spoiler function to show the correct answer choice - but the answer is 4 or choice E.

Where do feel there is a typo? I got this question from here:
https://www.urch.com/forums/gmat-problem ... -help.html

38n+3 + 2 is divided by 5?

It is not really CONCISE to write the question this way

38n+5 is more clear, and the remainder changes with different n values. So there is a typo

but if it were written as this:

38^(n+3) + 2

we will still get different remainders..

last option would be:

3^(8n+3)+2

AHA

3^(8n+3) will always have 7 at the unit digits

and we add 2 so the unit digit will always be 9

and divided by 5...

will ALWAYS give us 4

So bottom line is: TYPOS are waste of time, but we still can learn something while figuring out what they are!

LGTCH
---------------------
"DON'T LET ANYONE STEAL YOUR DREAM!"

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jnellaz: Master | Next Rank: 500 Posts; Posts: 114; Joined: Mon Oct 20, 2008 7:03 pmFollowed by: 5 members; Thanked: 4 times

by jnellaz » Fri Nov 07, 2008 10:53 am

Thanks for your post! How did you quickly figure out that 3^(8n+3) will always have 7 at the unit digits?

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logitech: Legendary Member; Posts: 2134; Joined: Mon Oct 20, 2008 11:26 pmFollowed by: 25 members; Thanked: 237 times; GMAT Score:730

by logitech » Fri Nov 07, 2008 10:58 am

jnellaz wrote:Thanks for your post! How did you quickly figure out that 3^(8n+3) will always have 7 at the unit digits?

the unit digit of power of three are 3,9,7, and 1 than they all repeat each other. Just play with the numbers and you will find out.

LGTCH
---------------------
"DON'T LET ANYONE STEAL YOUR DREAM!"

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jimmiejaz: Master | Next Rank: 500 Posts; Posts: 207; Joined: Sun Mar 11, 2007 6:16 pm; Location: Mumbai; Thanked: 11 times

by jimmiejaz » Fri Nov 07, 2008 11:01 am

jnellaz wrote:Thanks for your post! How did you quickly figure out that 3^(8n+3) will always have 7 at the unit digits?

Let me take this ahead logitech!!!!!
if you check 3 powers have the unit digit 3 after every 4 powers...
ie. 3^1=3 , 3^5=243
3^2=9, 3^6=729
3^3=27,3^7=2187
so, i hope u get the pattern now....
So put n =1, we get 3^11, n=2 we get 3^19 so.. the units digit will be the same as 3^3 or 7.
hope it helps....

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jnellaz: Master | Next Rank: 500 Posts; Posts: 114; Joined: Mon Oct 20, 2008 7:03 pmFollowed by: 5 members; Thanked: 4 times

by jnellaz » Fri Nov 07, 2008 11:13 am

Got it. Thanks to you both!

And for those of you out there that have read this and want a quick list of repeatable units for 2-12 powers here it is:

Some examples:

2^5 = 32 = Unit is 2.
5^12 = 244140625 =Unit is 5

2 = 2, 4, 8, 6
3 = 3, 9, 7, 1
4 = 4, 6
5 = 5
6 = 6
7 = 7, 9, 3, 1
8 = 8, 4, 2, 6
9 = 9, 1
10 = 0
11 = 1
12 = 2, 4,8, 6

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singalong: Senior | Next Rank: 100 Posts; Posts: 95; Joined: Wed Feb 09, 2011 6:42 amFollowed by: 1 member; Thanked: 1 times

by singalong » Sat Nov 19, 2011 1:22 am

logitech wrote:
jnellaz wrote:Thanks for your post! How did you quickly figure out that 3^(8n+3) will always have 7 at the unit digits?
the unit digit of power of three are 3,9,7, and 1 than they all repeat each other. Just play with the numbers and you will find out.

I am not able to connect the dots here.I understood that unit digit of power of three are 3, 9, 7 and 1.They form a cycle of 4.But how is 3^(8n+3) gonna have 7 at it's unit digit?

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8137; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

hi

by Scott@TargetTestPrep » Wed Dec 20, 2017 7:34 am

If n is a positive integer, what is the remainder when 38n+3 + 2 is divided by 5?

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

We need to determine the remainder of:

3^8n x 3^3 + 2 when it is divided by 5.

We only need to know the units digit of the above expression to determine the remainder.

Let's look at the pattern of units digits of powers of 3. Note that we are only concerned with the units digits, so, for example, for 3^3, we concern ourselves only with the units digit of 27, which is 7. Here is the pattern:

3^1 = 3

3^2 = 9

3^3 = 7

3^4 = 1

3^5 = 3

The repeating pattern is 3-9-7-1. Since the units digit pattern for a base of 3 is 3-9-7-1, we see that whenever 3 is raised to an exponent that is a multiple of 4, the units digit will be 1. Thus:

3^8n x 3^3 = units digit of 1 x units digit of 7 = units digit of 7

So, units digit of 7 + 2 = units digit of 9, and thus 9/5 has a remainder of 4.

Answer: E

Scott Woodbury-Stewart
Founder and CEO
[email protected]

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