Events & Promotions
|
|
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
Track
Your Progress
Practice
Pays
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Apr 2212:30 AM EDT
-01:30 AM EDT
At one point, she believed GMAT wasn’t for her. After scoring 595, self-doubt crept in and she questioned her potential. But instead of quitting, she made the right strategic changes. The result? A remarkable comeback to 695. Check out how Saakshi did it.
Apr 1612:00 PM EDT
-11:59 PM EDT
You’re first to know: Save $200 on GMAT prep starting tomorrow, 4/13. Get 6 official practice tests and study with real questions. Be ready to save!
Apr 2301:30 AM EDT
-02:30 AM EDT
Struggling with GMAT Verbal as a non-native speaker? Harsh improved his score from 595 to 695 in just 45 days—and scored a 99 %ile in Verbal (V88)! Learn how smart strategy, clarity, and guided prep helped him gain 100 points.
Apr 2308:00 PM PDT
-09:00 PM PDT
Video explanations + diagnosis of 10 weakest areas + 150+ short videos + study plan!
Apr 2610:00 AM EDT
-11:00 AM EDT
The Target Test Prep course represents a quantum leap forward in GMAT preparation, a radical reinterpretation of the way that students should study. Try before you buy with a 5-day, full-access trial of the course for FREE!
Apr 2711:00 AM EDT
-12:00 PM EDT
Prefer video-based learning? The Target Test Prep OnDemand course is a one-of-a-kind video masterclass featuring 400 hours of lecture-style teaching by Scott Woodbury-Stewart, founder of Target Test Prep and one of the most accomplished GMAT instructors
Apr 2808:00 AM PDT
-11:00 AM PDT
Whether you are just beginning your MBA journey or fine-tuning your path to a 700+ score, GMAT Day is designed to give you a competitive edge.
13 May 2019, 23:07
Kudos
Bookmarks
C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
75%
(hard)
Question Stats:
59% (02:04) correct
41%
(02:17)
wrong
based on 496
sessions
History
Date
Time
Result
Not Attempted Yet
When the number \(2^{1000}\) is divided by 13, the remainder in the division is
(A) 1
(B) 2
(C) 3
(D) 7
(E) 11
(A) 1
(B) 2
(C) 3
(D) 7
(E) 11
15 May 2019, 11:32
Kudos
Bookmarks
This problem is not the most usual type of remainder questions, tested on the GMAT. What makes this question special is, that, although it appears difficult and intimidating at first look, it can actually be solved using some very fundamental concepts on Remainders.
In any remainder question, play smart by trying to express the dividend in the form of (divisor + 1) or (divisor – 1) (as far as you can) so that the remainder turns out to be 1. Apart from this, use the following shortcuts to find out remainders for large values:
Remainder \(\frac{( A+ B )}{K}\) = Remainder \(\frac{(A)}{K}\) + Remainder \(\frac{(B)}{K}\)
Remainder \(\frac{(A * B)}{K}\) = Remainder \(\frac{(A)}{K}\) * Remainder \(\frac{(B)}{K}\)
The above concepts can be extended to the sum or product of any number of values.
Also, when you write the dividend (which has a power) in the form of \((divisor + 1) ^{power}\), the remainder will always be 1. However, if you write the same dividend as \((divisor – 1) ^ {power}\), the remainder will be 1 if the power is even and will be -1 if the power is odd. Since the remainder cannot be negative, we add the divisor to the negative remainder to obtain the final remainder.
Let us now try and apply these concepts to the problem at hand.
By writing down the first few powers of 2, we observe that \(2^6\) i.e. 64 is the number we can write in the form of (13k -1).
Therefore, \(2^{1000}\) can be written as (\(2^{996}\)) * (\(2^4\)). So,
Remainder \(\frac{(2^{1000})}{13}\) = Remainder \(\frac{(2^{996})}{13}\) * Rem\(\frac{(2^4)}{13}\).
Let’s now calculate the individual remainders.
Remainder \(\frac{(2^{996})}{13}\):
\(2^{996}\) = \((2^6) ^{166}\) = \((13*5 – 1)^{166}\).
So, here, we have written the dividend, \(2^{996}\) in the form of \((13k – 1) ^{even power}\), where 13 is the divisor. As discussed earlier, since the power is even, the remainder here will be 1.
Remainder\(\frac{(2^4)}{13}\) = Remainder\(\frac{(16)}{13}\)= 3.
Therefore, final remainder = 1 * 3 = 3.
So, the correct answer option is C.
It’s important to not get flustered by the sheer magnitude of the dividend. Because, as we have demonstrated, if you know the concepts that we have discussed in this post, this question is actually very straight forward. The only effort you will have to put in is to find out the power of 2 which can be written in the form of (13k -1) or (13k + 1). Once that is established, the rest of the solution is simple.
Hope that helps!
In any remainder question, play smart by trying to express the dividend in the form of (divisor + 1) or (divisor – 1) (as far as you can) so that the remainder turns out to be 1. Apart from this, use the following shortcuts to find out remainders for large values:
Remainder \(\frac{( A+ B )}{K}\) = Remainder \(\frac{(A)}{K}\) + Remainder \(\frac{(B)}{K}\)
Remainder \(\frac{(A * B)}{K}\) = Remainder \(\frac{(A)}{K}\) * Remainder \(\frac{(B)}{K}\)
The above concepts can be extended to the sum or product of any number of values.
Also, when you write the dividend (which has a power) in the form of \((divisor + 1) ^{power}\), the remainder will always be 1. However, if you write the same dividend as \((divisor – 1) ^ {power}\), the remainder will be 1 if the power is even and will be -1 if the power is odd. Since the remainder cannot be negative, we add the divisor to the negative remainder to obtain the final remainder.
Let us now try and apply these concepts to the problem at hand.
By writing down the first few powers of 2, we observe that \(2^6\) i.e. 64 is the number we can write in the form of (13k -1).
Therefore, \(2^{1000}\) can be written as (\(2^{996}\)) * (\(2^4\)). So,
Remainder \(\frac{(2^{1000})}{13}\) = Remainder \(\frac{(2^{996})}{13}\) * Rem\(\frac{(2^4)}{13}\).
Let’s now calculate the individual remainders.
Remainder \(\frac{(2^{996})}{13}\):
\(2^{996}\) = \((2^6) ^{166}\) = \((13*5 – 1)^{166}\).
So, here, we have written the dividend, \(2^{996}\) in the form of \((13k – 1) ^{even power}\), where 13 is the divisor. As discussed earlier, since the power is even, the remainder here will be 1.
Remainder\(\frac{(2^4)}{13}\) = Remainder\(\frac{(16)}{13}\)= 3.
Therefore, final remainder = 1 * 3 = 3.
So, the correct answer option is C.
It’s important to not get flustered by the sheer magnitude of the dividend. Because, as we have demonstrated, if you know the concepts that we have discussed in this post, this question is actually very straight forward. The only effort you will have to put in is to find out the power of 2 which can be written in the form of (13k -1) or (13k + 1). Once that is established, the rest of the solution is simple.
Hope that helps!
08 Jul 2020, 20:43
Kudos
Bookmarks
not sure if this is correct approach but
cyclicity of 2
2
4
8
6
2^1000 lands on 4. So Dividing 2^1000 by 13 should be the same remainder as with 2^4
2^4 = 16
16/13 = 1 reminder 3
Answer is 3
cyclicity of 2
2
4
8
6
2^1000 lands on 4. So Dividing 2^1000 by 13 should be the same remainder as with 2^4
2^4 = 16
16/13 = 1 reminder 3
Answer is 3
