Unit's digit of the product : GMAT Quantitative Section
Check GMAT Club App Tracker for the Latest School Decision Releases http://gmatclub.com/AppTrack

 It is currently 08 Dec 2016, 11:22

### 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

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# Unit's digit of the product

Author Message
TAGS:

### Hide Tags

Intern
Joined: 06 Jan 2013
Posts: 24
GPA: 3
WE: Engineering (Transportation)
Followers: 0

Kudos [?]: 51 [0], given: 9

What is the unit's digit [#permalink]

### Show Tags

06 May 2013, 11:14
What is the unit's digit of $$21^3$$x$$21^2$$x$$34^7$$x$$46^8$$x$$77^8$$?
A. 4
B. 8
C. 6
D. 2
E. 3
_________________

If you shut your door to all errors, truth will be shut out.

Intern
Joined: 06 Jan 2013
Posts: 24
GPA: 3
WE: Engineering (Transportation)
Followers: 0

Kudos [?]: 51 [0], given: 9

Unit's digit of the product [#permalink]

### Show Tags

06 May 2013, 11:20
If the unit's digit in the product (47n x 729 x 345 x 343) is 5, what is the maximum no. of values that n may take?
A. 9
B. 3
C. 7
D. 5
E. 4
_________________

If you shut your door to all errors, truth will be shut out.

VP
Status: Far, far away!
Joined: 02 Sep 2012
Posts: 1123
Location: Italy
Concentration: Finance, Entrepreneurship
GPA: 3.8
Followers: 180

Kudos [?]: 1911 [1] , given: 219

Re: What is the unit's digit [#permalink]

### Show Tags

06 May 2013, 11:24
1
KUDOS
The question can be solved using patterns: we have to find the unit digit of each term and them multiply them.

21 raised by any number will have 1 as last digit. ($$21*21=xx1$$ every time)

34 follows this pattern (I'll consider only the unit digit)
$$34^1=4$$
$$34^2=6$$
$$34^3=4$$
$$34^4=6$$
So $$34^7$$ (odd) $$= 4$$.

46 follows this pattern (every exp will end with a 6)
$$46^1=6$$
$$46^2=6$$
$$46^3=6$$
So $$46^8 = 6$$

77 follows this pattern
$$77^1=7$$
$$77^2=9$$
$$77^3=3$$
$$77^4=1$$ and then repeats
So $$77^8=1$$

Finally 1*1*4*6*1=24
_________________

It is beyond a doubt that all our knowledge that begins with experience.

Kant , Critique of Pure Reason

Tips and tricks: Inequalities , Mixture | Review: MGMAT workshop
Strategy: SmartGMAT v1.0 | Questions: Verbal challenge SC I-II- CR New SC set out !! , My Quant

Rules for Posting in the Verbal Forum - Rules for Posting in the Quant Forum[/size][/color][/b]

Intern
Joined: 06 Jan 2013
Posts: 24
GPA: 3
WE: Engineering (Transportation)
Followers: 0

Kudos [?]: 51 [0], given: 9

If a,b,c,d are consecutive odd numbers, then [#permalink]

### Show Tags

06 May 2013, 11:24
If a,b,c,d are consecutive odd numbers, then $$(a^2 + b^2 + c^2 + d^2)$$ is always divisible by:-
A. 5
B. 7
C. 3
D. 4
E. 6
_________________

If you shut your door to all errors, truth will be shut out.

Intern
Joined: 06 Jan 2013
Posts: 24
GPA: 3
WE: Engineering (Transportation)
Followers: 0

Kudos [?]: 51 [0], given: 9

When x is divided by 6, remainder obtained is 3. Find the [#permalink]

### Show Tags

06 May 2013, 11:27
When x is divided by 6, remainder obtained is 3. Find the remainder when $$x^4 + x^3 + x^2 + x + 1$$ is divided by 6.

A. 3
B. 4
C. 1
D. 5
E. 2
_________________

If you shut your door to all errors, truth will be shut out.

Last edited by GMATtracted on 06 May 2013, 11:53, edited 1 time in total.
VP
Status: Far, far away!
Joined: 02 Sep 2012
Posts: 1123
Location: Italy
Concentration: Finance, Entrepreneurship
GPA: 3.8
Followers: 180

Kudos [?]: 1911 [1] , given: 219

Re: Unit's digit of the product [#permalink]

### Show Tags

06 May 2013, 11:30
1
KUDOS
Consider only the last digits: $$9*5*3=135$$. Now 135*n must finish with a 5 (and n can be $$0,1,...,9$$)
135*0 will not end with a 5
135*1 will end with a 5 (5*1=5)
135*2 will not end with a 5 (5*2=10)

Only the ODD value of n mantain $$5$$ as the last digit, so n can be any odd value : $$1,3,5,7,9$$. Five possible values
_________________

It is beyond a doubt that all our knowledge that begins with experience.

Kant , Critique of Pure Reason

Tips and tricks: Inequalities , Mixture | Review: MGMAT workshop
Strategy: SmartGMAT v1.0 | Questions: Verbal challenge SC I-II- CR New SC set out !! , My Quant

Rules for Posting in the Verbal Forum - Rules for Posting in the Quant Forum[/size][/color][/b]

Intern
Joined: 06 Jan 2013
Posts: 24
GPA: 3
WE: Engineering (Transportation)
Followers: 0

Kudos [?]: 51 [1] , given: 9

Re: Unit's digit of the product [#permalink]

### Show Tags

06 May 2013, 11:55
1
KUDOS
Thanks Zarrolou for the explanation. This is my explanation:-
One very interesting thing about problems concerning unit's digit is that we can eliminate every other digit no matter how large the number is.
Thus, the expression would eventually reduce to finding the unit's digit for the expression:- $$1^3$$ x $$1^2$$ x $$4^7$$ x $$6^8$$ x $$7^8$$
Now the end digit for various indices of 4 can be written as:-
$$4^1$$------4
$$4^2$$------6
$$4^3$$------4
$$4^4$$------6
We clearly see that 4 and 6 are repeating in nature. Such a nature is called cyclicity and in case of number 4 the cyclicity is '2'
In the expression the power of 4 is 7 which is of the form $$(2n+1)$$.
Hence the unit's digit of $$4^7$$ is 4.
Similarly the cyclicity for nos. 6 and 7 can be written down as shown below:-
$$6^1$$-----6
$$6^2$$-----6
We observe that 6 to the power of any integer will eventually lead to a value having its unit digit 6.
Thus the unit's digit for $$6^8$$ would be 6.
For number 7:-
$$7^1$$-----7
$$7^2$$-----9
$$7^3$$-----3
$$7^4$$-----1
$$7^5$$-----7
Clearly then the cyclicity of 7 is 4.
And since the power of 7 in the expression is 8(which is of the form 4n) , therefore the unit's digit in this case would be 1
Thus, clearly the unit's digit of the resultant expression would be (1 x 1 x 4 x 6 x 1)=24.
_________________

If you shut your door to all errors, truth will be shut out.

VP
Status: Far, far away!
Joined: 02 Sep 2012
Posts: 1123
Location: Italy
Concentration: Finance, Entrepreneurship
GPA: 3.8
Followers: 180

Kudos [?]: 1911 [0], given: 219

Re: When x is divided by 6, remainder obtained is 3. Find the [#permalink]

### Show Tags

06 May 2013, 12:00
GMATtracted wrote:
When x is divided by 6, remainder obtained is 3. Find the remainder when $$x^4 + x^3 + x^2 + x + 1$$ is divided by 6.

A. 3
B. 4
C. 1
D. 5
E. 2

When x is divided by 6 the remainder is 3. Take x=3 (remaider of 3/6 is 3)

$$3^4 + 3^3 + 3^2 + 3 + 1$$.It requires a lil bit of old fashion math, but it never hurt anybody

$$81+27+9+3+1=121$$
Reminder of $$\frac{121}{6}$$ is 1

or:
$$3(27+9+3+1)+1=121$$
The first part is 3*(even>2) => divisible by 6. The remainder is 1

PS: your explanation of the above question works just fine
_________________

It is beyond a doubt that all our knowledge that begins with experience.

Kant , Critique of Pure Reason

Tips and tricks: Inequalities , Mixture | Review: MGMAT workshop
Strategy: SmartGMAT v1.0 | Questions: Verbal challenge SC I-II- CR New SC set out !! , My Quant

Rules for Posting in the Verbal Forum - Rules for Posting in the Quant Forum[/size][/color][/b]

VP
Status: Far, far away!
Joined: 02 Sep 2012
Posts: 1123
Location: Italy
Concentration: Finance, Entrepreneurship
GPA: 3.8
Followers: 180

Kudos [?]: 1911 [1] , given: 219

Re: If a,b,c,d are consecutive odd numbers, then [#permalink]

### Show Tags

06 May 2013, 12:15
1
KUDOS
GMATtracted wrote:
If a,b,c,d are consecutive odd numbers, then $$(a^2 + b^2 + c^2 + d^2)$$ is always divisible by:-
A. 5
B. 7
C. 3
D. 4
E. 6

Take "smart numbers":
a=-3 b=-1 c=1 d=3
20 is divisible by 5 and 4, two options remain

a=-1 b=1 c=3 d=5
36, is divisible by 3,4 and 6

So the only common divisor is 4
_________________

It is beyond a doubt that all our knowledge that begins with experience.

Kant , Critique of Pure Reason

Tips and tricks: Inequalities , Mixture | Review: MGMAT workshop
Strategy: SmartGMAT v1.0 | Questions: Verbal challenge SC I-II- CR New SC set out !! , My Quant

Rules for Posting in the Verbal Forum - Rules for Posting in the Quant Forum[/size][/color][/b]

Intern
Joined: 06 Jan 2013
Posts: 24
GPA: 3
WE: Engineering (Transportation)
Followers: 0

Kudos [?]: 51 [0], given: 9

Re: Unit's digit of the product [#permalink]

### Show Tags

07 May 2013, 05:14
Really nice explanation Zarrolou. Your method is the MOST suitable one for exams like GMAT. Taking examples of nos.(like 3 in this case) always helps reaching an answer faster.
Here is a more general explanation:-
From the question we can say that x will be of the form (6k+3), where k is an integer.
Therefore, the expression would be:-
$$(6k+3)^4 + (6k+3)^3 + (6k+3)^2 + (6k+3) + 1$$
What would the remainder be if the above expression is divided by 6?
If we expand each bracketed expression given above, every individual term of the expanded expression would contain '6k' except the following:-
$$3^4, 3^3, 3^2, 3 and 1$$
So, the question reduces to----->"What would the remainder be if $$(3^4+3^3+3^2+3+1)$$ is divided by 6?"
Now, if we observe carefully any power of 3($$like 3^2, 3^3, 3^4.....or 3^{1000} as the case may be$$) when divided by 6 always leaves a remainder 3.
Therefore, $$(3^4+3^3+3^2+3+1)$$ can be written as:-(6m+3)+(6n+3)+(6p+3)+(6q+3)+1=6(m+n+p+q+2)+1.Hence 1 is the required remainder
_________________

If you shut your door to all errors, truth will be shut out.

Re: Unit's digit of the product   [#permalink] 07 May 2013, 05:14
Similar topics Replies Last post
Similar
Topics:
7 To find the units digit of a large number? 8 03 Jul 2013, 21:58
units digit 11 19 Jul 2011, 11:55
Unit digit of a series. 2 31 Jul 2010, 13:50
2 Units Digit 8 22 Apr 2010, 06:16
What's the units digit Questions? 5 31 Mar 2008, 16:09
Display posts from previous: Sort by