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
we will pick new questions that match your level based on your Timer History
Track Your Progress
every week, we’ll send you an estimated GRE score based on your performance
Practice Pays
we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:
Your score will improve and your results will be more realistic
Is there something wrong with our timer?Let us know!
Targeting Round 1? Then join Sia Admissions founder, Susan Berishaj, in this R1-focused session on March 28th, where she will guide you on the importance of a plan for a successful application cycle. Register now and simplify your application journey
Join Brian and many other students who have used Target Test Prep to score high on the GRE. Start your 5-day trial of the TTP GRE Course today for FREE.
Re: What is the remainder when
[#permalink]
30 Jan 2021, 09:54
1
Hi,
This can also be solved using cyclicity. Cyclicity of any number is about the last digit and how they appear in a certain defined manner. Let's take an example to clear this thing: The cyclicity chart of 2 is: 2^1 =2. 2^2 =4. 2^3= 8. 2^4=16 2^5=32. Thus, 2 has a cyclicity of 4. Hence to solve this question we need 2^16 and we only need the units digit to find the remainder when divided by 10. Now consider 2^16, we need to divide 16 by 4 why because 2 has a cyclicity of 4 thus we are left with no remainder and therefore 2^16 has a units digit similar to 2^4=6. Similarly 3 and 7 has a cyclicity of 4 and both 3^4 and 7^4 has units digit 1.
Thus 6x1x1=6. 6/10 leaves us with remainder 6
IMO C
Note: If we were asked something like 2^14 then, we will divide 14 by 4 and we are left with remainder 2. And then count the digits from 2^1=2 whose units digit is also equal to 2^13 and 2^2 whose units digit is also equal to 2^14 thus the units digit of 2^2 will also be the units digit of 2^14 ==4
Now, we have split 42 into two numbers, one (40) is a number closer to 42 and a multiple of 10 and other is a small number
Now, if we expand this using Binomial theorem then we will get all terms except the last term as a multiple of 40 => A multiple of 10
=> All terms except the last term will give us a remainder of 0 when divided by 10
=> Remainder of \((2^{16})(3^{16})(7^{16})\) by 10 is same as remainder of the last term = 16C16 * 2^16 * 40^0 = 2^16 by 10
Theory: Remainder of a number by 10 is same as remainder of the unit's digit of that number by 10
Now, Let's find the unit's digit of \(2^{16}\) first.
We can do this by finding the pattern / cycle of unit's digit of power of 2 and then generalizing it.
Unit's digit of \(2^1\) = 2 Unit's digit of \(2^2\) = 4 Unit's digit of \(2^3\) = 8 Unit's digit of \(2^4\) = 6 Unit's digit of \(2^5\) = 2
So, unit's digit of power of 2 repeats after every \(4^{th}\) number. => We need to divided 16 by 4 and check what is the remainder => 16 divided by 4 gives 0 remainder
=> \(2^{16}\) will have the same unit's digit as \(2^4\) = 6 => Unit's digits of \(2^{16}\) = 6
But remainder of \(2^{16}\) by 10 = 6
So, Answer will be D Hope it helps!
Learn How to Find Remainders with 2, 3, 5, 9, 10 and Binomial Theorem