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 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 2512:30 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 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.
Updated on: 13 Aug 2018, 03:11
Originally posted by EgmatQuantExpert on 09 Apr 2015, 22:45.
Last edited by EgmatQuantExpert on 13 Aug 2018, 03:11, edited 7 times in total.
Last edited by EgmatQuantExpert on 13 Aug 2018, 03:11, edited 7 times in total.
Kudos
Bookmarks
E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
25% (03:03) correct
75%
(03:13)
wrong
based on 1554
sessions
History
Date
Time
Result
Not Attempted Yet
P and Q are prime numbers less than 70. What is the units digit of P*Q?
(1) Units’ digit of \((P^{4k+2} - Q\)) is equal to 7, where k is a positive integer.
(2) Units digit of the expression \([PQ + Q*(Q+1) - Q^2]\) is a perfect cube
This is
Register for our Free Session on Number Properties (held every 3rd week) to solve exciting 700+ Level Questions in a classroom environment under the real-time guidance of our Experts
(1) Units’ digit of \((P^{4k+2} - Q\)) is equal to 7, where k is a positive integer.
(2) Units digit of the expression \([PQ + Q*(Q+1) - Q^2]\) is a perfect cube
This is
Ques 8 of The E-GMAT Number Properties Knockout
Register for our Free Session on Number Properties (held every 3rd week) to solve exciting 700+ Level Questions in a classroom environment under the real-time guidance of our Experts
Updated on: 07 Aug 2018, 03:07
Originally posted by EgmatQuantExpert on 11 Apr 2015, 02:19.
Last edited by EgmatQuantExpert on 07 Aug 2018, 03:07, edited 3 times in total.
Last edited by EgmatQuantExpert on 07 Aug 2018, 03:07, edited 3 times in total.
Kudos
Bookmarks
Detailed Solution
Step-I: Given Info
We are given two prime numbers \(P, Q < 70\). We are asked to find the units digit of \(P*Q\).
Step-II: Interpreting the Question Statement
The unit’s digit of the product of \(P\) & \(Q\) would depend on the units digit of \(P\) & \(Q\). Hence, our endeavour would be to find the units digit of \(P\) & \(Q\).
Step-III: Statement-I
Statement-I tells us that the difference between two numbers is odd which would imply opposite even/odd natures of the numbers i.e. one is even and the other is odd. Since, there is only on even number (i.e. \(2\)), there are two possible scenarios:
• \(P=2\), so \(P ^{4k+2}\) will have the units digit as \(4\). As a result the units digit of \(Q\) would be \(7\). Hence units digit of product of \(P\) & \(Q\) would be \(4\)
• \(Q=2\), in such a case units digit of \(P^{4k+2}\) will be \(9\), which implies that units digit of \(P\) is either \(7\) or \(3\). Hence, units digit of product of \(P\) & \(Q\) would be either \(4\) or \(6\).
Since we do not have a unique answer, Statement-I is not sufficient to answer the question.
Step-IV: Statement-II
The expression in statement-II can be simplified to \(Q(P+1)\), since its unit digit is a perfect cube, the possible values are \(1\) and \(8\).
If unit digit is \(1\),
• Units digit of \(P\)= \(2\) and units digit of \(Q\) =\(7\), so the units digit of product of \(P\) & \(Q\) would be \(4\)
If unit digit is \(8\),two cases are possible:
• Units digit of \(Q= 2\) and units digit of \(P =3\), so the units digit of product of \(P\) & \(Q\) would be \(6\)
• Units digit of \(Q= 1\) and units digit of \(P= 7\), so the units digit of product of \(P\) & \(Q\) would be \(7\)
Since we do not have a unique answer, Statement-II is not sufficient to answer the question
Step-V: Combining Statements I & II
Statement-I tells us that the units digit of \(P\) & \(Q\) can be \(4\) or \(6\). Statement-II tells us that the units digit of \(P\) & \(Q\) can be \(4\) or \(6\) or \(7\).
By Combining statements- I & II, we still do not have a unique answer.
Thus combination of St-I & II is also insufficient to answer the question.
Hence, the correct answer is Option E
Key Takeaways
1. Know the properties of Even-Odd combinations to save the time spent deriving them in the test
2. In even-odd questions, simplify complex expressions into simpler expressions using the properties of even-odd combinations
3. Know the cyclicity of the numbers to arrive at their units digit
Regards
Harsh
Step-I: Given Info
We are given two prime numbers \(P, Q < 70\). We are asked to find the units digit of \(P*Q\).
Step-II: Interpreting the Question Statement
The unit’s digit of the product of \(P\) & \(Q\) would depend on the units digit of \(P\) & \(Q\). Hence, our endeavour would be to find the units digit of \(P\) & \(Q\).
Step-III: Statement-I
Statement-I tells us that the difference between two numbers is odd which would imply opposite even/odd natures of the numbers i.e. one is even and the other is odd. Since, there is only on even number (i.e. \(2\)), there are two possible scenarios:
• \(P=2\), so \(P ^{4k+2}\) will have the units digit as \(4\). As a result the units digit of \(Q\) would be \(7\). Hence units digit of product of \(P\) & \(Q\) would be \(4\)
• \(Q=2\), in such a case units digit of \(P^{4k+2}\) will be \(9\), which implies that units digit of \(P\) is either \(7\) or \(3\). Hence, units digit of product of \(P\) & \(Q\) would be either \(4\) or \(6\).
Since we do not have a unique answer, Statement-I is not sufficient to answer the question.
Step-IV: Statement-II
The expression in statement-II can be simplified to \(Q(P+1)\), since its unit digit is a perfect cube, the possible values are \(1\) and \(8\).
If unit digit is \(1\),
• Units digit of \(P\)= \(2\) and units digit of \(Q\) =\(7\), so the units digit of product of \(P\) & \(Q\) would be \(4\)
If unit digit is \(8\),two cases are possible:
• Units digit of \(Q= 2\) and units digit of \(P =3\), so the units digit of product of \(P\) & \(Q\) would be \(6\)
• Units digit of \(Q= 1\) and units digit of \(P= 7\), so the units digit of product of \(P\) & \(Q\) would be \(7\)
Since we do not have a unique answer, Statement-II is not sufficient to answer the question
Step-V: Combining Statements I & II
Statement-I tells us that the units digit of \(P\) & \(Q\) can be \(4\) or \(6\). Statement-II tells us that the units digit of \(P\) & \(Q\) can be \(4\) or \(6\) or \(7\).
By Combining statements- I & II, we still do not have a unique answer.
Thus combination of St-I & II is also insufficient to answer the question.
Hence, the correct answer is Option E
Key Takeaways
1. Know the properties of Even-Odd combinations to save the time spent deriving them in the test
2. In even-odd questions, simplify complex expressions into simpler expressions using the properties of even-odd combinations
3. Know the cyclicity of the numbers to arrive at their units digit
Regards
Harsh
EgmatQuantExpert
(1) Units’ digit of \((P^{4k+2} - Q\)) is equal to 7, where k is a positive integer.
(2) Units digit of the expression \([PQ + Q*(Q+1) - Q^2]\) is a perfect cube
Perfect squares are integers: 0, 1, 4, 9, 16 and
Similarly, perfect cubes are: 0, 1, 8, 27, 64.
(1) Units’ digit of \((P^{4k+2} - Q\)) is equal to 7, where k is a positive integer.
\((P^{4k+2} - Q\)) = \((P^{4k}*P^{2} - Q\))
as the unit digit is ODD (7) either P or Q is 2 (that's the only even prime) .
IF P is EVEN and Q is ODD PRIME
if P=2 , we know that 2 has a cyclicity of 4 , so , \((P^{4k}*P^{2} )\) will always have a unit digit 4.
how ? \((P^{4k}\) will have a unit digit of 6 and \(P^{2}\) will have a unit digit of 4 and \(6*4 = 24\).
now Q's unit has to be 7.
Q Can be 7, 17, 37 67
Unit digit is 4 in this case .
lets look at other case when Q=2 and P is some ODD prime.
explained below: -
\((P^{4k}*P^{2} - Q\))
\((P^{4k}*P^{2} - 2\))
IF P=3 Q=2 the above expression has a unit digit of 7 . unit digit of P*Q= 6
IF P=7 Q=2 the above expression has unit digit of 7. unit digit of P*Q= 4
This option is not sufficient .
(2) Units digit of the expression \([PQ + Q*(Q+1) - Q^2]\) is a perfect cube
this is one is easy ...
\([PQ + Q*(Q+1) - Q^2]\)
=\([PQ + Q]\)
=\([Q(P + 1)]\)
the above product should result into a unit digit of 0,1,8
P=19, Q=31 unit digit 0
P=2, Q=7 unit digit 1
P=7, Q=11 unit digit 8
Not Sufficient.
Together:
we know that Either P or Q is Even=2, as per statement A.
If P=2 , Q Can be 7, 17, 37 67
from stmt B
IF P=3 Q=2 the above expression has a unit digit of 7 .
IF P=7 Q=2 the above expression has unit digit of 7.
and \([Q(P + 1)]\) has a unit digit which is perfect cube (0,1,8) .
if P=2 , P+1=3 , Q can be 7,17,37,67 . unit digit of PQ will be 4 .
but if Q=2 , P has a unit digit of 3 or 7 such that the product \([Q(P + 1)]\) will not have a unit digit which is perfect cube (0,1,8) .
answer C
