|
|
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.
26 May 2017, 04:42
Kudos
Bookmarks
If \(a\) and \(b\) are positive integers, what is the remainder when \(7^{4a} + b\) is divided by \(4\)?
(1) \(a = 2\)
(2) \(b = 3\)
(1) \(a = 2\)
(2) \(b = 3\)
26 May 2017, 04:42
Kudos
Bookmarks
Official Solution:
For this question, you need two preliminary knowledge.
First preliminary knowledge is that the units digit of \(7^{n}\) has the cycle of
\(7 \rightarrow 9 \rightarrow 3 \rightarrow 1 \rightarrow 7 \rightarrow 9 \rightarrow 3 \rightarrow \ldots\) ,
and the tens digit of \(7^{n}\) has the cycle of \(0 \rightarrow 4 \rightarrow 4 \rightarrow 0 \rightarrow 0 \rightarrow 4 \rightarrow 4 \rightarrow \ldots\)
In other words, \(7^{n}\) is
\(n=1\) \(n=2\) \(n=3\) \(n=4\) \(n=5\) \(n=6\) \(n=7\)
\(07 \rightarrow \sim49 \rightarrow \sim43 \rightarrow \sim01 \rightarrow \sim07 \rightarrow \sim49 \rightarrow \sim43 \rightarrow \ldots\) . As such, since n is the exponent of \(7^{n}\), it has the cycle of 4th exponent.
Second preliminary knowledge is that the remainder of \(n\) divided by \(4\) is equal to the remainder of dividing unit's digits and ten's digits of \(n\) by \(4\). This is because \(100 = 4*25\), and digits above the hundreds digit is all comprised of multiples of \(100\), so you can divide the numbers by \(4\) just to the tens digit.
Coming back to the original question, you can take the 1st step of the variable approach and modify the original condition and the question.
Since the remainder of \(7^{4a}+b\) divided by \(4\) is equal to the remainder of dividing unit's digits and ten's digits of \(7^{4a}+b\) by \(4\), you get \(7^{4a}+b=(7^{4})^{a}+b=(\sim01)+b\), and therefore you only need to know what \(b\) is. According to con 2), \(b=3\), and so you always get \(7^{4a}+3=(\sim01)+3=\sim04\). Also, since you only need to know that the remainder of this number divided by \(4\) is \(\sim04, 04\) has the remainder of \(0\) when divided by \(4\), hence unique, and sufficient. Therefore, the answer is B.
Answer: B
For this question, you need two preliminary knowledge.
First preliminary knowledge is that the units digit of \(7^{n}\) has the cycle of
\(7 \rightarrow 9 \rightarrow 3 \rightarrow 1 \rightarrow 7 \rightarrow 9 \rightarrow 3 \rightarrow \ldots\) ,
and the tens digit of \(7^{n}\) has the cycle of \(0 \rightarrow 4 \rightarrow 4 \rightarrow 0 \rightarrow 0 \rightarrow 4 \rightarrow 4 \rightarrow \ldots\)
In other words, \(7^{n}\) is
\(n=1\) \(n=2\) \(n=3\) \(n=4\) \(n=5\) \(n=6\) \(n=7\)
\(07 \rightarrow \sim49 \rightarrow \sim43 \rightarrow \sim01 \rightarrow \sim07 \rightarrow \sim49 \rightarrow \sim43 \rightarrow \ldots\) . As such, since n is the exponent of \(7^{n}\), it has the cycle of 4th exponent.
Second preliminary knowledge is that the remainder of \(n\) divided by \(4\) is equal to the remainder of dividing unit's digits and ten's digits of \(n\) by \(4\). This is because \(100 = 4*25\), and digits above the hundreds digit is all comprised of multiples of \(100\), so you can divide the numbers by \(4\) just to the tens digit.
Coming back to the original question, you can take the 1st step of the variable approach and modify the original condition and the question.
Since the remainder of \(7^{4a}+b\) divided by \(4\) is equal to the remainder of dividing unit's digits and ten's digits of \(7^{4a}+b\) by \(4\), you get \(7^{4a}+b=(7^{4})^{a}+b=(\sim01)+b\), and therefore you only need to know what \(b\) is. According to con 2), \(b=3\), and so you always get \(7^{4a}+3=(\sim01)+3=\sim04\). Also, since you only need to know that the remainder of this number divided by \(4\) is \(\sim04, 04\) has the remainder of \(0\) when divided by \(4\), hence unique, and sufficient. Therefore, the answer is B.
Answer: B
Moderator:
