Aptitude - Numbers - Discussion

Discussion :: Numbers - General Questions (Q.No.10)

10. 

What is the unit digit in {(6374)1793 x (625)317 x (341491)}?

[A].
[B]. 2
[C]. 3
[D]. 5

Answer: Option A

Explanation:

Unit digit in (6374)1793 = Unit digit in (4)1793

    = Unit digit in [(42)896 x 4]

    = Unit digit in (6 x 4) = 4

Unit digit in (625)317 = Unit digit in (5)317 = 5

Unit digit in (341)491 = Unit digit in (1)491 = 1

Required digit = Unit digit in (4 x 5 x 1) = 0.


Hemanth said: (Aug 23, 2010)  
This can be done in a simpler manner.

The unit digit of 625^x is 5.

The unit digit of 6374^y is an even number.

When multiplied, the unit digit must be 0.

Vaibhav said: (Sep 13, 2010)  
Thanks.

A.Athiban Sakthivel said: (Nov 2, 2010)  
How unit digit in [(4^2)896 x 4] is equal to Unit digit in (6 x 4) = 4 ?

Supreet Singh said: (Nov 18, 2010)  
I too. !

But hemanth method is good.

Hanah said: (Nov 21, 2010)  
@Hemanth I still can't understand your method.

Don said: (Dec 2, 2010)  
What happend? jus can't understand. Hemanth method is good but what if it is 622^ () then wat?

Anitha said: (Dec 6, 2010)  
I can't understand.

Sangee said: (Dec 8, 2010)  
Please tel me how you got (6*4).

Rather than your idea is superb!

Revathi said: (Dec 8, 2010)  
I cant understand... explain (6*4)....

Jhansi said: (Dec 29, 2010)  
Because 4^2=16, unit digit is 6....
like that
4^4=256, unit digit is 6..
in 4^6 , unit digit is 6......

so in (4^2)896 unit digit is 6 because 896 is an even number..

Hope you all understand..

Seema said: (Jan 21, 2011)  
Seema:As 6374 = unit digit is 4,
in 625 = itz 5 n in 341 = itz 1
just multiply all the degit units = 4*5*1 = 20
in 20 = unit digit is 0
So, ans is 0

Priyankavanga said: (Feb 2, 2011)  
@Seema

I am satisfied with your answer.

Thanku very much.

Suhas said: (Feb 12, 2011)  
Great discussion to understand the unit digit.

Karthi said: (Feb 21, 2011)  
What is Unit digit ?

Nandish said: (Mar 8, 2011)  
Thank you seema. Got it.

Venkatesh said: (Apr 19, 2011)  
Find the unit digit in (264)^102 + (264)^102.

Venkatesh said: (Apr 19, 2011)  
Find the unit digit in (264)^102 + (264)^103

Shahid said: (May 19, 2011)  
The process generally is like this

Say 4^0=1 5^0=1 1^0=1 (1 time)
4^1=4 5^1=5 1^1=1 (2nd time)
4^2=16 5^2=25 1^2=1 (3rd time)
4^3=64 5^3=625 1^4=1 (4th time)

So as per problem units number must be the result of
(4^1793)*(5^317)*(1^491)

From the above calculations done it is evident that
1^anything =1 and 5^any positive number gives 5 as units digit
Similarly with 4 powers

Now take 4^1793.......First divide the power 1793/4 gives remainder 1.

So Units digit is (4^1)*5*1=20 whose
units digit is 0

Emela said: (Jun 9, 2011)  
How can you divide 1793 simply by 4 and write that remainder? is it applicable for any number like 6^, 7^.

Pratiksha said: (Jun 29, 2011)  
What is unit digit ?

Shubhi said: (Jul 3, 2011)  
The unit digit is simply another term for the ones place..
the unit digit in 2343546 is 6
the unit digit in 234 is 4.. etc :)

Goutham said: (Jul 4, 2011)  
To find unit digit in a large powered number, simply consider its last digit only. for suppose in (6374)^1793 take 4^1793 only. now note that any odd power of 4 gives a number with its unit digit as 4.so unit digit in (6374)^1793 is '4'.and then any power of 5 gives a number with 5 as its unit digit.so u.d in (625)^317 is '5'.then any power of 1 gives only 1 itself. so u.d in 341^491 is '1'. finally multiply all numbers i.e., 4*5*1 which gives 20 with u.d as 0.hence that result.

Satchi Mishra said: (Jul 6, 2011)  
Remember, all digits at the power of 5 gives the value having unit place containing the original digit. Special digits (0,1,5,6) raised to any power always give the value having unit place containing the original digit.

In this, 1793/5= ### and remainder is 3,
Unit value of 6374 is to be raised to power of 3 = 4^3= Unit value is 4.

Similarly 317/5= ### and remainder is 2,
Unit value of 625 is to be raised to power of 2 = 5^2= Unit value is 5.

Last factor comes in special digit i.e. 1 and will always give 1 at unit place,if raised to any power.

Hence, the unit place of the result will be 4x5x1= 20; 0 at unit place.

Vasanthi said: (Jul 15, 2011)  
@Karthi
Unit digit means last digit.

Gaurav said: (Jul 23, 2011)  
Seema is right. I like her method. The way of approaching is good. Toooooo much.

Shashank said: (Sep 20, 2011)  
@gautham. You made it very simple!

Thanks.

Dharmraj said: (Oct 16, 2011)  
Why are you consider (5) the power 317 is digit unit is 5? please explain it.

Swetha said: (Nov 4, 2011)  
Seema 's method is too good. It saves time. Thank you seema.

Aruna said: (Dec 1, 2011)  
Unit digit means last digit..
5^(anything) is ending with 5..

So for our convenience it s considered as 5.

Abc said: (Dec 17, 2011)  
This can be done in a simpler manner.

The unit digit of 625^x is 5.

The unit digit of 6374^y is an even number.

When multiplied, the unit digit must be 0.

How?

Sasi said: (Dec 27, 2011)  
When finally mutilpled (4*5*1) we get 20. Consider the 20, the last digit ill be the unit digit. So the answer is 0,

Got it.

Prabagaran said: (Jan 21, 2012)  
No. How to separate this sequence [(42)896 x 4].

Dibakara Nayajk said: (Feb 22, 2012)  
4* 4 = 16 - 6 ->2
16 *4 = 84 - 4 ->3

5 * n = ..5
1 8 n = 1
(6 /4 ) * 5 * 1 = ......0


so ans is 0;

Deepi said: (Mar 19, 2012)  
Thanks to seema and jhansi.

Abhishek said: (Mar 25, 2012)  
Hemant method is very good..
Thx

Ramya said: (Jan 22, 2013)  
Its so simple,

Unit digit of 6374 is 4
Unit digit of 625 is 5
Unit digit of 341 is 1
Just multiply 4*5*1=20
So unit digit of 20 is 0

Answer is 0.

Dr R Vasudevan said: (Jan 25, 2013)  
For a number ending in 4 ( like 6534) any odd power will have unit digit 4 and Even power unit digit 6 ( 1731 is odd and hence 6534 6 1731 will have unit digit 4.

For a number ending in 5 ( like 625 ) any odd or even power , unit digit will result in 5

For a number ending in 1 ( like 341 ) any odd or even power , unit digit will result in 1

4*5*1 = 20 - Unit digit 0.

Kishore said: (Mar 19, 2013)  
For finding total result is there any logic? could you please explain?

Ananya said: (Apr 2, 2013)  
If the unit digit of (x-1)^2 is 5 and (x+1)^2 is 9 then determine the units of x^2?

Nil said: (Apr 9, 2013)  
@Satchi Mishra.

You are wrong. Not applicable to few no.s like 2^9.
2^9 = 512. unit digit 2.

By your method:

9/5=1, remainder 4.
2^4= 16. unit digit 6.

Not matches the case.

Vasundhara .G said: (Aug 2, 2013)  
how can you make XI (11) as nine (9) just using a single line using only once
you must not replace or erase any line ...just draw a line any where to make it give nine

Raja Shekhar said: (Sep 28, 2013)  
What is unit digit? please explain anybody.

Rajkumar said: (Oct 28, 2013)  
How unit digit of (6374)^1793 is 4?

Please explain.

Saurabh said: (Nov 20, 2013)  
If we have to find tens digit then how we find and the expression is same { (6374) ^1793 x (625) ^317 x (341^491) }?

Vicky said: (Dec 7, 2013)  
6374 U.D is 4.
625 U.D is 5.

341 U.D is 1.
4*5*1=20.

20 U.D is 0.
Answer is zero.

Deepak Chaudhari said: (Jan 10, 2014)  
Look Friends its Simple:

Just take the last digit of number and last digit of power.

1). Here given is 6374^1793 means you should take 4^3 which is 64
means for 4^3 (4*4=16 and 16*4 is 64 or 6*4 is 24 whose unit digit is 4(remember for finding unit digit always take last digit after multiplication).

2). 625^317 means 5^7 here remember any power to 5 will always have 5 as its unit digit. So here the no is 5.

3). 341^491 means 1^1 is always 1 in its unit digit.

So from 1, 2, and 3 we got three numbers ie 4, 5, 1.

Now multiply these numbers and take the last digit of the product which will be your answer(4*5*1=20).

So 0 is your answer.

Manas Singh said: (Jan 25, 2014)  
What is the unit digit of 7^63* 3^57 * 6^41 ?

Daniel Jeff said: (Feb 4, 2014)  
What mean by unit digit?

Arun said: (Mar 17, 2014)  
As well say for this: 31.999*12.001*17.5001.

Nisha said: (Aug 8, 2014)  
What is a unit digit? please anyone explain me.

Kratika said: (Aug 8, 2014)  
Find unit digit of 1^1!+2^2!+........+80^80.

Amit Kumar Singh said: (Aug 9, 2014)  
On the basis of cyclicity rule we easily find the unit digit .

4*5*1 = 0.
4^1 =4 5^1 =5 1^1=1.
4^2 =6 5^2 =5 1 cyclicity is 1.
4^3 =4 5 cyclicity is 5.
Cyclicity of 4 is 2.

Then 4^1793 divide 1793/3 = 1 4^1 = 4 as on...

Finally 4*5*1 = 0.

Vinay said: (Oct 15, 2014)  
I have for the given problem.

4*5*1= 20 that is unit digit =0;

But for (264)^102*(264)^103 we need to get answer as 4*4 = 16 unit digit as 6,

But in test book they given answer as 0,

How it is please explain me?

Hidd said: (Nov 13, 2014)  
When power is same we can just multiple the unit number, we will get answer.

Shivani said: (Dec 3, 2014)  
Thanks @Seema. I got your point otherwise all of them make me confuse.

Susmitha said: (Dec 13, 2014)  
What is the unit digit of {3^65*6^59*7^71}? Please can anyone explain me the answer?

Vidhya said: (Jan 9, 2015)  
How to find easy way to answer the question?

Abhignareddy said: (Feb 1, 2015)  
(2467)^153*(341)^72 solve this please.

Neeha said: (Feb 7, 2015)  
Please explain in briefly.

Suryasish said: (Feb 24, 2015)  
@Abhigyan reddy.

The units digit can be 1, 3, 7 or 9.

Praveen said: (Apr 1, 2015)  
Power are not equals then what we do.

Shubham said: (Apr 7, 2015)  
Hi I have few questions.

1) In the above problem we have taken the unit digits of every number regardless of of the power.

2) If any such type questions come can we use the same method to solve the problem. Please answer.

Jeevitha said: (May 4, 2015)  
Given question is.

{(6374)^1793 x (625)^317 x (341)^491}.

Ans:

NOTE: Unit digit for (4)^odd num is 4.

Unit digit for (4)^even num is 6.
Unit digit for (5)^any num is 5.
Unit digit for (1)^any num is 1.

Now apply the above conditions,

Unit digit of 6374 is 4.
Unit digit of 625 is 5.
Unit digit of 341 is 1.

Just multiply 4*5*1 = 20.

So unit digit of 20 is 0.

Answer is 0.

Varun said: (Jun 20, 2015)  
What will be the remainder if 2^856 is divided by 9?

Can any one mail me shortest trick?

Dinesh Sen said: (Jun 22, 2015)  
7 is multiplied by 7 107 times, unit place number is?

Dr. Pratik Sharma said: (Jun 25, 2015)  
7*7^107.
i.e 7^108.

We know 7^1 unit place will be 7.

7^2 unit place will be 9.
7^3 unit place will be 3.
7^4 unit place will be 1.

So unit place will be 7, 9, 3, 1 after these 7, 9, 3, 1 again and again.

We have 108 that is divisible by 4 so unit place will be one. You can find it on scientific calculator as well.

Prasanna Kartik said: (Jul 8, 2015)  
Hi guys,

These type of problems are solved by cyclicity rule.

Here this is the problem.

6374)^1793x(625)^317x(341)^491.

First term's unit digit is 4 cylicity for 4 is 4 (4^1) , 6 (4^2) , 4 (4^3) , 6 (4^4). This cycle continues in the same pattern. (Remember I only consider unit digits). So cyclicity is 2.

Now check first term is power 1793 is divisible by 2 or not. Clearly it shows that it will leave reminder 1. Which means unit digit is 4 for first term.

Second term unit digit 5, and last term is 1 so the answer is 4X5X1=20. Unit digit is 0. Note: Make a table for cylicity for numbers 1 to 9. Then it will become very handy.

Venugopal said: (Aug 15, 2015)  
Thanks. I understood the answer. But when the end no is 3. E.g. (2463) power 323.

Then how to solve explain this?

Nanditha said: (Sep 7, 2015)  
Answer for 2463^323 may be after considering the cyclicity of 3 i.e., 4 we may get 3 as the answer.

Anu said: (Sep 13, 2015)  
Find unit digit of 1^1+2^2+3^3+4^4+5^5+6^6?

Neelam said: (Sep 14, 2015)  
4^2 = 16, 4^3 = 64. Unit digit = 4, when power is 3 (odd).

Unit digit of any number having odd power remains same as the original number has;here 6374^1793. Power is odd so unit digit will be 4, similarly 625^317 & 341^491 unit digits are 5 and 1 respectively, since power is odd.

So the unit digit of overall expression = (4.5.1) = 20 i.e. 0.

Bipin said: (Sep 21, 2015)  
(6374)^1793.

= Here unit digit is 4 then we can write (4)^1793.

As we know that multiple of 4 will be = 4, 16, 64, 256 = Here unit digit is 4 and 6 continuing coming then we have to take 2 nos. which should come 4 and 6 that's why we took.

= (4)^1793.

= 4*(4)^1792 (we can write it).

= 4*(4^2)^896.

Take only unit digit = 4*16 = 4*6 = 24 = 4.

Same as:

Unit digit in (625) 317 = Unit digit in (5) 317 = 5.

Unit digit in (341) 491 = Unit digit in (1) 491 = 1.

Answer:

Required digit = Unit digit in (4 x 5 x 1) = 0.

Vivek said: (Nov 22, 2015)  
Follow this Rule.

Unit Digit of Base, Powers, Unit digit of product.

0, 1, 5, 6, any powers, same digit.

3, 7, 9, power exactly divisible by 4, 1.

2, 4, 8 " 6.

So in this problem: {(6374)^1793x(625)^317x(341)^491)}.

= (6374)^1793 unit digit is 4.

Apply the above rule: Divide 1793 by 4, we get remainder as 1.

Take the remainder to the power of unit digit's i.e. 4^1.

So we get (6374)^1793 = 4. For (625) ^317 same digit as the unit digit.

So we get (625)^317 = 5. For (341)^(491) same digit as the unit digit.

So we get (341)^(491) = 1.

Finally (4*5*1) = 20.

Thus unit digit is 0 = Answer.

Shital said: (Dec 4, 2015)  
(209*144) ^2+ (209*209) + (209*144) + (144*144) =?

Please tell me.

Sai Chandana said: (Jun 29, 2016)  
@Hemanth.

Your explanation was clear and good. Thanks!

Amrutha P Nair said: (Jul 4, 2016)  
Friends. There is a simple way to get this answer.

Consider the unit digit means the last digit. Here in this question, 4 is the unit digit of 6374 and 5 is the unit digit of 625 and 1 is the unit digit of 341.

We want to find the unit digit. So avoid the other numbers.

So, the next step is a simple method we only using the logic behind the multiplication.

(4)^any odd no = 4.
(4)^any even no = 6.
(5)^any no = 5.
(1) any no =1 what I meant that.

4^1 = 4.
4^2 = 16.
4^3 = 64.

i.e, the unit digit are 4 & 6 and this is repeating. Did you get my point?

So by using this method.

(4) ^1793 * (5) ^317 * (1) ^491 = ?
4 * 5 * 1 = 20.

So the unit digit is 0.

Hope, this help you. Thank you.

Karishma said: (Jul 20, 2016)  
896 * 2 = 1792.....?

Bhavani said: (Jul 20, 2016)  
Thank you @Seema.

Renu said: (Jul 22, 2016)  
Thank you @Seema.

Ragh said: (Jul 30, 2016)  
Find the product of the first 8 terms of a 15 number sequence whose 1st, 3rd, 5th numbers are 60, 45 and 30?

Radhakrishnan E said: (Aug 3, 2016)  
yes you are good @Seema.

Sravanthi said: (Aug 22, 2016)  
@Seema, you have done a good job.

Nataraj said: (Aug 26, 2016)  
Please give the clarity to solve the problem.

Harikrishna said: (Sep 17, 2016)  
Your method is very easy. Thank you @Ramya.

Nidhin M Z said: (Sep 30, 2016)  
Unit Digit of Base, Powers, Unit digit of the product.

0, 1, 5, 6, any powers, same digit.
3, 7, 9, power exactly divisible by 4, 1.
2, 4, 8 " 6.

So in this problem: {(6374)^1793 x (625)^317 x (341)^491)}.
= (6374)^1793 unit digit is 4.

Apply the above rule: Divide 1793 by 4, we get remainder as 1.

Take the remainder of the power of unit digit's i.e. 4^1.

So we get (6374)^1793 = 4. For (625) ^317 same digit as the unit digit.
So we get (625)^317 = 5. For (341)^(491) the same digit as the unit digit.
So we get (341)^(491) = 1.
Finally (4 * 5 * 1) = 20.

Thus unit digit is 0 = Answer.

Apurva Raj said: (Nov 23, 2016)  
Anybody can find the remainder of this question (5^625) /7?

Reshma said: (Dec 13, 2016)  
Thank you @Nidhin M Z.

Hemant said: (Jan 2, 2017)  
How - (5)^317 = 5?

I solved - (5^2)^158 * 5 it gives unit digit 0.

Sowjanya said: (May 19, 2017)  
There is simply way to get this answer.

{(6374)1793*(625)317*(314 )491)}.
Unit place is the last number.
4 * 5 * 1 = 20.
Last digit is 0.

Teju said: (Jul 25, 2017)  
Thanks @Sowjanya.

Raj said: (Aug 4, 2017)  
Thanks @Jhansi.

Harshal said: (Jan 17, 2018)  
In simpler manner for 4^x.
For,x=Odd unit place is 4.
and x=Even unit place is 6.

Sabu said: (Jul 22, 2018)  
Understand the answer, Thank you, @Seema.

Himanshi said: (Aug 7, 2018)  
Consider this:

For,x=Odd, the unit place for 4^x is 4. The unit place for 5^x is 5, The unit place for 1^x is 1.
and x=Even, the unit place for 4^x is 6. The unit place for 5^x is 0, The unit place for 1^x is 1.

=>unit place means the digit present at the one's place(rightmost digit) of a number.

Ahankar said: (Aug 14, 2018)  
Simply, We have to take last digit of every number so by taking last number as 4, 5, 1. Now multiply these 4*5*1=20 so now take last digit of answer so it is 0 (zero).

Arpitha M said: (Oct 16, 2018)  
In short method,

6374^1793.
1793/4=448, remainder 01.
So answer is 4^1(take the last digit in 6374).

In 625^325.
325/4=79remainder 01.
So answer is 5^1.

In 341^491.
491/4=122remainder 3.
So 1^3.

Finally;
4^1*5^1*1^3=20.
In 20, take last digit 0 is the answer.

Hola said: (Nov 20, 2018)  
Simply, We have to take last digit of every number so by taking last number as 4, 5, 1. Now multiply these 4*5*1=20 so now take last digit of answer so it is 0 (zero).

Mrunal said: (Nov 22, 2018)  
Thanks @Hola.

Nafeesa said: (Dec 18, 2018)  
Thanks for the answer @Vasudevan.

Keerthinayak said: (Feb 24, 2019)  
Thanks @Seema.

Nilkanta Das said: (Mar 19, 2019)  
Nice explanation, Thanks all.

Shivam Rana said: (Mar 26, 2019)  
Thanks @Seema.

Pooja said: (Apr 1, 2019)  
As in 6374 = unit digit is '4'
in 625 = unit digit is '5'
in 341 = unit digit is '1'.

Now just multiply all the digit units =4 * 5 * 1=20
In 20, the unit digit is '0'.

Again,
In the power of 6374 i.e,1793 = unit digit is '3',
In the power of 625 i.e, 317 = unit digit is '7',
In the power of 341 i.e, 491 = unit digit is '1',
Now multiply all the digit units = 3*7*1=21.
In 21, the unit digit is '1'.
Hence,
The required unit digit is, 0^1.
= 0 (answer).

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