Find the remainder of 2^100/12

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Vincen: Legendary Member; Posts: 2898; Joined: Thu Sep 07, 2017 2:49 pmFollowed by: 5 members; Thanked: 6 times

by Vincen » Mon Jan 29, 2018 3:14 am

Find the remainder of 2^100/12

A. 4
B. 2
C. 8
D. 1
E. none

The OA is the option A.

Is there a simple way to solve this PS question? I can I solve it? I'd appreciate your help.

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Source: Beat The GMAT — Problem Solving | Mon Jan 29, 2018 3:14 am

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Mon Jan 29, 2018 4:21 am

Vincen wrote:Find the remainder of 2^100/12

A. 4
B. 2
C. 8
D. 1
E. none

Test easy exponents and look for a pattern.

2Â²/12 = 4/12 = 0 R4.
2Â³/12 = 8/12 = 0 R8.
2â�´/12 = 16/12 = 1 R4.
2â�µ/12 = 32/12 = 2 R8.
2â�¶/12 = 64/12 = 5 R4.
2â�·/12 = 128/12 = 10 R8.

The cases above illustrate the following:
When the exponent is EVEN, the remainder is 4.
When the exponent is ODD, the remainder is 8.
Since 2Â¹â�°â�°/12 has an even exponent, the remainder will be 4.

The correct answer is A.

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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

by Scott@TargetTestPrep » Tue Jan 30, 2018 10:32 am

Vincen wrote:Find the remainder of 2^100/12

A. 4
B. 2
C. 8
D. 1
E. none

Let's find a remainder pattern:

2^1/12 = 0 remainder 2

2^2/12 = 0 remainder 4

2^3/12 = 0 remainder 8

2^4/12 = 1 remainder 4

2^5/12 = 2 remainder 8

2^6/12 = 5 remainder 4

We see that, besides 2^1/12, we have this pattern: when 2 is raised to an even power, the remainder is 4, and when 2 is raised to an odd power the remainder is 8. Thus, 2^100/12 has a reminder of 4.

Answer: A

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EconomistGMATTutor: GMAT Instructor; Posts: 555; Joined: Wed Oct 04, 2017 4:18 pmFollowed by: 12 members; Thanked: 180 times

by EconomistGMATTutor » Wed Jan 31, 2018 1:11 pm

Find the remainder of 2^100/12

A. 4
B. 2
C. 8
D. 1
E. none

The OA is the option A.

Is there a simple way to solve this PS question? I can I solve it? I'd appreciate your help.

Hi Vincen,
Let's take a look at your question.

We need to find the remainder of
$$\frac{2^{100}}{12}$$
$$=\frac{2^{2\times50}}{12}$$
$$=\frac{4^{50}}{12}$$

Let's check what happens when powers of 4 is divided by 12, starting from 4^2.
$$4^2=16$$
$$\frac{16}{12}=1\ remainder\ 4$$

$$4^3=64$$
$$\frac{64}{12}=5\ remainder\ 4$$

$$4^4=256$$
$$\frac{256}{12}=21\ remainder\ 4$$

$$4^5=1024$$
$$\frac{1024}{12}=85\ remainder\ 4$$

Which shows that remainder is 4 for all the positive powers of 4.
Therefore, option A is correct.

Hope it helps.
I am available if you'd like any follow up.

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