If n is a positive integer, what is the remainder

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

If n is a positive integer, what is the remainder

by Gmat_mission » Sun Jun 24, 2018 11:55 pm

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If n is a positive integer, what is the remainder when 3^(8n+3) + 2 is divided by 5?

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

[spoiler]OA=E[/spoiler]

Is there a fast way to solve this PS question? Could someone explain this to me? Thanks in advance.

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Source: Beat The GMAT — Problem Solving | Sun Jun 24, 2018 11:55 pm

Jeff@TargetTestPrep: GMAT Instructor; Posts: 1462; Joined: Thu Apr 09, 2015 9:34 am; Location: New York, NY; Thanked: 39 times; Followed by:22 members

by Jeff@TargetTestPrep » Tue Jun 26, 2018 5:20 pm

Gmat_mission wrote:If n is a positive integer, what is the remainder when 3^(8n+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

Jeffrey Miller
Head of GMAT Instruction
[email protected]