GMAT Question of the Day - Daily to your Mailbox; hard ones only

It is currently 14 Oct 2019, 21:29

Close

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

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 GMAT 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.

Close

Request Expert Reply

Confirm Cancel

N is a two-digit number, whose tens digit is 9. If the units digit of

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Find Similar Topics 
e-GMAT Representative
User avatar
V
Joined: 04 Jan 2015
Posts: 3074
N is a two-digit number, whose tens digit is 9. If the units digit of  [#permalink]

Show Tags

New post Updated on: 16 Dec 2016, 05:32
1
7
00:00
A
B
C
D
E

Difficulty:

  75% (hard)

Question Stats:

64% (02:39) correct 36% (02:27) wrong based on 122 sessions

HideShow timer Statistics

Take a stab at the following GMAT like question on Units Digit. The official solution will be posted soon.

N is a two-digit number, whose tens digit is 9. If the units digit of \(N^p\) and \(N^q\) are 6 and 4 respectively. Find N.

    (1) P is a positive even integer.
    (2) Q is a positive odd integer.


Answer Options
    A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
    B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
    C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
    D. EACH statement ALONE is sufficient to answer the question asked.
    E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.


To practise ten 700+ Level Number Properties Questions attempt the The E-GMAT Number Properties Knockout

.Click here.

Image

_________________

Originally posted by EgmatQuantExpert on 16 Dec 2016, 04:01.
Last edited by EgmatQuantExpert on 16 Dec 2016, 05:32, edited 5 times in total.
e-GMAT Representative
User avatar
V
Joined: 04 Jan 2015
Posts: 3074
N is a two-digit number, whose tens digit is 9. If the units digit of  [#permalink]

Show Tags

New post Updated on: 25 Dec 2016, 07:49
Steps 1 & 2: Understand Question and Draw Inferences
Given: The tens digit of N is 9 and the unit’s digit of Np and Nq are 6 and 4 respectively.

To find: The value of N

Let us consider \(N = AB\), where A and B are single-digit numbers and the value of A is 9

Therefore, \(N^p = (AB)^p\)

To find the units digit, we need to focus on the last digit of the number, which is B in the above case. Hence, we can write \(B^p= 6\)

The above expression will hold true in the following cases –

Case I - Value of B = 2, Value of p = 4m
Case II - Value of B = 4, Value of p = 2m
Case III - Value of B = 6, Value of p = any number
Case IV - Value of B = 8, Value of p = 4m

From the above cases, we can infer that finding the value of B on the basis of the value of p is possible only when p is an odd number because –

If p is a multiple of 4, B can take values - 2,4,6 or 8
If p is a multiple of 2 but not a multiple of 4 then B can be either 4 or 6
If p is an odd number, B can take only one value - 6

Using the same logic, we can write \(N^q\) as \((AB)6^q\)  and \(B^q = 4\).
The above expression will hold true in the following cases –

Case I - Value of B = 2, Value of q = 4m+2
Case II - Value of B = 4, Value of q = 2m+1
Case III - Value of B = 8, Value of q = 4m+2

From the above cases, we can infer that finding the value of B on the basis of the value of q is only possible when q is an odd number because –

If q is an even number, B can be 2 or 8
If q is an odd number, B has to be 4


Step 3: Analyze Statement 1 independently
Statement 1 says “P is a positive even integer”.

From the inferences that we have drawn, we know if P is an even number, the units digit of N can be 2, 4, or 8

Since Statement 1 leads us to 4 possible values of B, it is clearly not sufficient to arrive at a unique answer.

Step 4: Analyze Statement 2 independently

Statement 2 says that Q is a positive odd integer.

From the inferences that we have drawn, we know if Q is an odd number, the units digit of N has to be 4

Thus N = 94.

Statement 2 is sufficient to get us a unique answer.

Step 5: Analyze Both Statements Together (if needed)

Since we’ve already got a unique answer in Step 4, this step is not required

Hence Correct Answer: Option B
_________________

Originally posted by EgmatQuantExpert on 16 Dec 2016, 04:02.
Last edited by EgmatQuantExpert on 25 Dec 2016, 07:49, edited 1 time in total.
Intern
Intern
User avatar
Joined: 15 Nov 2015
Posts: 25
Location: Netherlands
Schools: HEC Dec '17 (II)
GMAT 1: 730 Q47 V44
GPA: 3.99
Re: N is a two-digit number, whose tens digit is 9. If the units digit of  [#permalink]

Show Tags

New post 16 Dec 2016, 06:00
Let N = ab and a = 9. Now N = 9b. We need to find b to find N.

Given: \(N^p = 6\) and \(N^q = 4\)

So N needs to have unit digits 6 and 4 if raised to certain powers.
This gives three options:
b = 2, 4 or 8.
For example:
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 6

1) p = even. Out of the three options, 2^4 (even) = 6 and 4^2 (even) = 6 and 8^4 (even) = 6. So insufficient.

2) q = odd. Out of the three options, there is only one b that produces unit digit 4 when raised to an odd power:
2 raised to an odd power alternatives between 2 and 8.
8 raised to an odd power alternatives between 8 and 2.
4 raised to an odd power is always 4.

So statement 2 is sufficient and answer is B.
e-GMAT Representative
User avatar
V
Joined: 04 Jan 2015
Posts: 3074
Re: N is a two-digit number, whose tens digit is 9. If the units digit of  [#permalink]

Show Tags

New post 16 Dec 2016, 06:21
koenh wrote:
Let N = ab and a = 9. Now N = 9b. We need to find b to find N.

Given: \(N^p = 6\) and \(N^q = 4\)

So N needs to have unit digits 6 and 4 if raised to certain powers.
This gives three options:
b = 2, 4 or 8.
For example:
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 6

1) p = even. Out of the three options, 2^4 (even) = 6 and 4^2 (even) = 6 and 8^4 (even) = 6. So insufficient.

2) q = odd. Out of the three options, there is only one b that produces unit digit 4 when raised to an odd power:
2 raised to an odd power alternatives between 2 and 8.
8 raised to an odd power alternatives between 8 and 2.
4 raised to an odd power is always 4.

So statement 2 is sufficient and answer is B.


Hi,

Good analysis. You've solved the question correctly and you've taken into account all possible cases while solving it. The best part of your solution is that you've done a thorough Question Statement analysis before moving on to analyze the Statements (1) and (2) independently. Doing so reduces the chances of making an error while solving DS questions.

Good job :)

Regards,
Saquib
_________________
e-GMAT Representative
User avatar
V
Joined: 04 Jan 2015
Posts: 3074
Re: N is a two-digit number, whose tens digit is 9. If the units digit of  [#permalink]

Show Tags

New post 25 Dec 2016, 07:49
GMATH Teacher
User avatar
P
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 935
N is a two-digit number, whose tens digit is 9. If the units digit of  [#permalink]

Show Tags

New post 28 Oct 2018, 15:08
EgmatQuantExpert wrote:
N is a positive two-digit number, whose tens digit is 9. If the units digit of \(N^p\) and \(N^q\) are 6 and 4 respectively. Find N.

    (1) P is a positive even integer.
    (2) Q is a positive odd integer.

\(\left\langle N \right\rangle = {\rm{units}}\,\,{\rm{digit}}\,\,{\rm{of}}\,\,N\,\,\,\)

\(\left. \begin{gathered}
N\,\, = \,\,\underline 9 \,\underline a \,\, \hfill \\
\left\langle {{N^p}} \right\rangle = 6\,\,\, \hfill \\
\left\langle {{N^q}} \right\rangle = 4 \hfill \\
\end{gathered} \right\}\,\,\,\,\,? = N\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\boxed{\,\,? = a\,\,}\)

\(\left( 1 \right)\,\,\,p \ge 2\,\,\,{\rm{even}}\,\,\,\left\{ \matrix{
\,\,\left\langle {{{92}^4}} \right\rangle = 6\,\,\,\,{\rm{and}}\,\,\,\,\,\left\langle {{{92}^2}} \right\rangle = 4\,\,\,\,\,\, \Rightarrow \,\,\,\,\,a = 2\,\,\,\,{\rm{viable}}\,\, \hfill \cr
\,\,\left\langle {{{94}^2}} \right\rangle = 6\,\,\,\,{\rm{and}}\,\,\,\,\,\left\langle {{{94}^1}} \right\rangle = 4\,\,\,\,\,\, \Rightarrow \,\,\,\,\,a = 4\,\,\,{\rm{viable}}\,\, \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{INSUFF}}.\,\)

\(\left( 2 \right)\,\,\,\left\{ \begin{gathered}
\,q \geqslant 1\,\,\,{\text{odd}} \hfill \\
\,\left\langle {{N^q}} \right\rangle = 4 \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\mathop \Rightarrow \limits^{{\text{inspection}}} \,\,\,\,\,a = 4\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\text{SUFF}}{\text{.}}\)

Obs.: inspection means: if a = 0,1,2,3,5,6,7,8 or 9, when a is put to an odd positive power, we don´t get units digit equal to 4, as needed.


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
GMAT Club Bot
N is a two-digit number, whose tens digit is 9. If the units digit of   [#permalink] 28 Oct 2018, 15:08
Display posts from previous: Sort by

N is a two-digit number, whose tens digit is 9. If the units digit of

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  





Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne