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

 It is currently 16 Jul 2018, 13:34

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

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

If a and b are distinct positive integers, what is the units digit of

Author Message
TAGS:

Hide Tags

Intern
Joined: 20 Aug 2010
Posts: 34
If a and b are distinct positive integers, what is the units digit of [#permalink]

Show Tags

Updated on: 08 May 2015, 08:58
6
00:00

Difficulty:

75% (hard)

Question Stats:

52% (01:59) correct 48% (02:07) wrong based on 135 sessions

HideShow timer Statistics

If a and b are distinct positive integers, what is the units digit of 2^a*8^b*4^(a+b)?

(1) b = 24 and a < 24
(2) The greatest common factor of a and b is 12

Originally posted by Postal on 26 Nov 2011, 02:13.
Last edited by Bunuel on 08 May 2015, 08:58, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
e-GMAT Representative
Joined: 04 Jan 2015
Posts: 1755
Re: If a and b are distinct positive integers, what is the units digit of [#permalink]

Show Tags

28 May 2015, 02:59
4
1
sagarag wrote:
I am still not able to get why it is B.. request someone help in getting the same understand...

Hi sagarag,

The question here is testing about the knowledge of cyclicity of the number 2. We know that cyclicity of 2 is 4 i.e. the unit's digit of 2 would repeat itself after every 4 powers. For example:

$$2^1 -> 2$$
$$2^2 -> 4$$
$$2^3 -> 8$$
$$2^4 -> 16$$
$$2^5 -> 32$$. Unit's digit of $$2^1$$ and $$2^5$$ is the same, i.e. they repeat after every 4 powers

Since the question prompt can be simplified to $$2^{3a +5b}$$, we just need to find what values can $$3a + 5b$$ take.

St-II tells us that GCD(a, b) =12. As 12 is the factor of both $$a$$ and $$b$$ we can write $$a$$ and $$b$$ as $$12x$$ and $$12y$$ respectively.

So $$3a + 5b = 12(3x +5y) = 4(9x + 15y)$$. This expression is always divisible by 4. So we can write $$2^{3a + 5b} = 2^{4n}$$ where $$n = 9x + 15y$$

We know that $$2^4$$, $$2^8$$, $$2^{12}$$.....$$2^{4n}$$ have the same units digit as 6.

Since st-II tells us that $$3a +5b$$ is a multiple of 4, we can definitely find the unit's digit of the expression $$2^{3a +5b}$$

Hope this helps

Regards
Harsh
_________________

Ace GMAT quant
Articles and Question to reach Q51 | Question of the week

Number Properties – Even Odd | LCM GCD
Word Problems – Percentage 1 | Percentage 2 | Time and Work 1 | Time and Work 2 | Time, Speed and Distance 1 | Time, Speed and Distance 2
Advanced Topics- Permutation and Combination 1 | Permutation and Combination 2 | Permutation and Combination 3 | Probability
Geometry- Triangles 1 | Triangles 2 | Triangles 3 | Common Mistakes in Geometry
Algebra- Wavy line

Practice Questions
Number Properties 1 | Number Properties 2 | Algebra 1 | Geometry | Prime Numbers | Absolute value equations | Sets

| '4 out of Top 5' Instructors on gmatclub | 70 point improvement guarantee | www.e-gmat.com

General Discussion
VP
Joined: 24 Jul 2011
Posts: 1439
GMAT 1: 780 Q51 V48
GRE 1: Q800 V740
Re: If a and b are distinct positive integers, what is the units digit of [#permalink]

Show Tags

26 Nov 2011, 02:45
1
1
The given expression can be simplified to 2^(a+3b+2a+2b) = 2^(3a+5b)

Using statement (1), the units digit will change depending on the value of a. Insufficient.
Using statement (2), both a and b have to be multiples of 12, which means that 3a+5b will always come out to be a multiple of 4. Using cyclicity, we can then say that the units digit of 2^(3a+5b) = units digit of 2^4 = 6. Sufficient.

(B) it is.
_________________

GyanOne | Top MBA Rankings and MBA Admissions Blog

Premium MBA Essay Review|Best MBA Interview Preparation|Exclusive GMAT coaching

Get a FREE Detailed MBA Profile Evaluation | Call us now +91 98998 31738

Intern
Joined: 13 May 2014
Posts: 26
GMAT Date: 11-01-2014
If a and b are distinct positive integers, what is the units digit of [#permalink]

Show Tags

28 May 2015, 00:24
I am still not able to get why it is B.. request someone help in getting the same understand...
Current Student
Joined: 14 Oct 2013
Posts: 45
Re: If a and b are distinct positive integers, what is the units digit of [#permalink]

Show Tags

28 May 2015, 08:00
Can you explain this portion of your explanation....

So 3a+5b=12(3x+5y)=4(9x+15y). This expression is always divisible by 4. So we can write 2^3a+5b=2^4n where n=9x+15y

Couldn't 12(3x+5y) also be = 6(6x+10y) so then the expression is always divisible by 6 so we can rewrite as 2^3a+5b=2^6n which wouldnt that give us a different unit digit than 2^4n?

Thanks for clarifying!
Intern
Joined: 06 Nov 2013
Posts: 10
Concentration: Technology, General Management
Re: If a and b are distinct positive integers, what is the units digit of [#permalink]

Show Tags

28 May 2015, 21:28
EgmatQuantExpert wrote:
sagarag wrote:
I am still not able to get why it is B.. request someone help in getting the same understand...

Hi

The question here is testing about the knowledge of cyclicity of the number 2. We know that cyclicity of 2 is 4 i.e. the unit's digit of 2 would repeat itself after every 4 powers. For example:

$$2^1 -> 2$$
$$2^2 -> 4$$
$$2^3 -> 8$$
$$2^4 -> 16$$
$$2^5 -> 32$$. Unit's digit of $$2^1$$ and $$2^5$$ is the same, i.e. they repeat after every 4 powers

Since the question prompt can be simplified to $$2^{3a +5b}$$, we just need to find what values can $$3a + 5b$$ take.

St-II tells us that GCD(a, b) =12. As 12 is the factor of both $$a$$ and $$b$$ we can write $$a$$ and $$b$$ as $$12x$$ and $$12y$$ respectively.

So $$3a + 5b = 12(3x +5y) = 4(9x + 15y)$$. This expression is always divisible by 4. So we can write $$2^{3a + 5b} = 2^{4n}$$ where $$n = 9x + 15y$$

We know that $$2^4$$, $$2^8$$, $$2^{12}$$.....$$2^{4n}$$ have the same units digit as 6.

Since st-II tells us that $$3a +5b$$ is a multiple of 4, we can definitely find the unit's digit of the expression $$2^{3a +5b}$$

Hope this helps

Regards
Harsh

Can you please explain this part above :-

So $$3a + 5b = 12(3x +5y) = 4(9x + 15y)$$. This expression is always divisible by 4. So we can write $$2^{3a + 5b} = 2^{4n}$$ where $$n = 9x + 15y$$

We know that $$2^4$$, $$2^8$$, $$2^{12}$$.....$$2^{4n}$$ have the same units digit as 6.
e-GMAT Representative
Joined: 04 Jan 2015
Posts: 1755
Re: If a and b are distinct positive integers, what is the units digit of [#permalink]

Show Tags

28 May 2015, 21:45
1
healthjunkie wrote:
Can you explain this portion of your explanation....

So 3a+5b=12(3x+5y)=4(9x+15y). This expression is always divisible by 4. So we can write 2^3a+5b=2^4n where n=9x+15y

Couldn't 12(3x+5y) also be = 6(6x+10y) so then the expression is always divisible by 6 so we can rewrite as 2^3a+5b=2^6n which wouldnt that give us a different unit digit than 2^4n?

Thanks for clarifying!

Hi healthjunkie,

Let's assume an example of a number $$2^{12}$$ for which we need to find its units digit. $$2^{12} = 2^6 * 2^6$$. We know that cyclicity of 2 is 4, so $$2^6$$ will have the same units digit as $$2^2$$ which is 4. So, units digit of $$2^{12} = 4 * 4$$ which will give us 6.

For a number $$2^x$$, units digit is 6 when $$x$$ is a multiple of 4. It does not matter to us if $$x$$ is also a multiple of 6, 12, 24 or any other number greater than 4. We can express all of them with 4 as the base. So

$$2^5 = 2^{4 +1} =$$ Units digit same as $$2^1$$
$$2^6 = 2^{4+2} =$$ Units digit same as $$2^2$$
$$2^7 = 2^{4 +3} =$$ Units digit same as $$2^3$$
$$2^8 = 2^{4 + 4} =$$ Units digit same as $$2^4$$
$$2^9 = 2^{2*4 +1} =$$ Units digit same as $$2^1$$
and the same pattern continues

Given the cyclicity of a number, try to express the power of the number in terms of the cyclicity of the number to find its units digit. In other words assume power as the dividend, cyclicity as the divisor, the remainder will decide the units digit of the expression.

Hope it's clear

Regards
Harsh
_________________

Ace GMAT quant
Articles and Question to reach Q51 | Question of the week

Number Properties – Even Odd | LCM GCD
Word Problems – Percentage 1 | Percentage 2 | Time and Work 1 | Time and Work 2 | Time, Speed and Distance 1 | Time, Speed and Distance 2
Advanced Topics- Permutation and Combination 1 | Permutation and Combination 2 | Permutation and Combination 3 | Probability
Geometry- Triangles 1 | Triangles 2 | Triangles 3 | Common Mistakes in Geometry
Algebra- Wavy line

Practice Questions
Number Properties 1 | Number Properties 2 | Algebra 1 | Geometry | Prime Numbers | Absolute value equations | Sets

| '4 out of Top 5' Instructors on gmatclub | 70 point improvement guarantee | www.e-gmat.com

e-GMAT Representative
Joined: 04 Jan 2015
Posts: 1755
Re: If a and b are distinct positive integers, what is the units digit of [#permalink]

Show Tags

28 May 2015, 21:50
EgmatQuantExpert wrote:
sagarag wrote:
I am still not able to get why it is B.. request someone help in getting the same understand...

Hi

The question here is testing about the knowledge of cyclicity of the number 2. We know that cyclicity of 2 is 4 i.e. the unit's digit of 2 would repeat itself after every 4 powers. For example:

$$2^1 -> 2$$
$$2^2 -> 4$$
$$2^3 -> 8$$
$$2^4 -> 16$$
$$2^5 -> 32$$. Unit's digit of $$2^1$$ and $$2^5$$ is the same, i.e. they repeat after every 4 powers

Since the question prompt can be simplified to $$2^{3a +5b}$$, we just need to find what values can $$3a + 5b$$ take.

St-II tells us that GCD(a, b) =12. As 12 is the factor of both $$a$$ and $$b$$ we can write $$a$$ and $$b$$ as $$12x$$ and $$12y$$ respectively.

So $$3a + 5b = 12(3x +5y) = 4(9x + 15y)$$. This expression is always divisible by 4. So we can write $$2^{3a + 5b} = 2^{4n}$$ where $$n = 9x + 15y$$

We know that $$2^4$$, $$2^8$$, $$2^{12}$$.....$$2^{4n}$$ have the same units digit as 6.

Since st-II tells us that $$3a +5b$$ is a multiple of 4, we can definitely find the unit's digit of the expression $$2^{3a +5b}$$

Hope this helps

Regards
Harsh

Can you please explain this part above :-

So $$3a + 5b = 12(3x +5y) = 4(9x + 15y)$$. This expression is always divisible by 4. So we can write $$2^{3a + 5b} = 2^{4n}$$ where $$n = 9x + 15y$$

We know that $$2^4$$, $$2^8$$, $$2^{12}$$.....$$2^{4n}$$ have the same units digit as 6.

Please go through the above post and let me know if you need clarification on any other point.

Hope this helps

Regards
Harsh
_________________

Ace GMAT quant
Articles and Question to reach Q51 | Question of the week

Number Properties – Even Odd | LCM GCD
Word Problems – Percentage 1 | Percentage 2 | Time and Work 1 | Time and Work 2 | Time, Speed and Distance 1 | Time, Speed and Distance 2
Advanced Topics- Permutation and Combination 1 | Permutation and Combination 2 | Permutation and Combination 3 | Probability
Geometry- Triangles 1 | Triangles 2 | Triangles 3 | Common Mistakes in Geometry
Algebra- Wavy line

Practice Questions
Number Properties 1 | Number Properties 2 | Algebra 1 | Geometry | Prime Numbers | Absolute value equations | Sets

| '4 out of Top 5' Instructors on gmatclub | 70 point improvement guarantee | www.e-gmat.com

Director
Joined: 12 Nov 2016
Posts: 774
Location: United States
Schools: Yale '18
GMAT 1: 650 Q43 V37
GRE 1: Q157 V158
GPA: 2.66
Re: If a and b are distinct positive integers, what is the units digit of [#permalink]

Show Tags

27 Sep 2017, 07:44
Postal wrote:
If a and b are distinct positive integers, what is the units digit of 2^a*8^b*4^(a+b)?

(1) b = 24 and a < 24
(2) The greatest common factor of a and b is 12

This question certainly takes some drilling- multiples of 2 have units digit that follow the cycle 2 4 8 6 - we can also rewrite and simplify the stimulus before going into the statements- a must

2^(4b + 4a)

Statement 1

Several possibilities

insuff

Statement 2

Although we cannot identify a or b it doesn't matter because we know they are positive integers and if they are factors of 12 then the only possibilities are 1 and 12, 2 and 6, 4 and 3. If we go back and plug these numbers in we will have three different sums: 28, 32, 52; however, these numbers are all divisible by 4 - and if we understand how to find the units digit of a number then we can clearly see that with all these combinations the units digit of 2^(4b + 4a) is bound to be 6 Bunuel am I right?

B
Intern
Joined: 30 Aug 2017
Posts: 18
Location: India
Schools: ISB '19 (A)
GMAT 1: 670 Q47 V35
GPA: 3.9
Re: If a and b are distinct positive integers, what is the units digit of [#permalink]

Show Tags

30 Sep 2017, 02:48
EgmatQuantExpert wrote:
sagarag wrote:
I am still not able to get why it is B.. request someone help in getting the same understand...

Hi sagarag,

The question here is testing about the knowledge of cyclicity of the number 2. We know that cyclicity of 2 is 4 i.e. the unit's digit of 2 would repeat itself after every 4 powers. For example:

$$2^1 -> 2$$
$$2^2 -> 4$$
$$2^3 -> 8$$
$$2^4 -> 16$$
$$2^5 -> 32$$. Unit's digit of $$2^1$$ and $$2^5$$ is the same, i.e. they repeat after every 4 powers

Since the question prompt can be simplified to $$2^{3a +5b}$$, we just need to find what values can $$3a + 5b$$ take.

St-II tells us that GCD(a, b) =12. As 12 is the factor of both $$a$$ and $$b$$ we can write $$a$$ and $$b$$ as $$12x$$ and $$12y$$ respectively.

So $$3a + 5b = 12(3x +5y) = 4(9x + 15y)$$. This expression is always divisible by 4. So we can write $$2^{3a + 5b} = 2^{4n}$$ where $$n = 9x + 15y$$

We know that $$2^4$$, $$2^8$$, $$2^{12}$$.....$$2^{4n}$$ have the same units digit as 6.

Since st-II tells us that $$3a +5b$$ is a multiple of 4, we can definitely find the unit's digit of the expression $$2^{3a +5b}$$

Hope this helps

Regards
Harsh

Can we take 3 common and apply the divisibility rule of 3?
Re: If a and b are distinct positive integers, what is the units digit of   [#permalink] 30 Sep 2017, 02:48
Display posts from previous: Sort by

Events & Promotions

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.