Aptitude - Problems on H.C.F and L.C.M - Discussion

Discussion :: Problems on H.C.F and L.C.M - General Questions (Q.No.14)

14. 

The least number which when divided by 5, 6 , 7 and 8 leaves a remainder 3, but when divided by 9 leaves no remainder, is:

[A]. 1677
[B]. 1683
[C]. 2523
[D]. 3363

Answer: Option B

Explanation:

L.C.M. of 5, 6, 7, 8 = 840.

Required number is of the form 840k + 3

Least value of k for which (840k + 3) is divisible by 9 is k = 2.

Required number = (840 x 2 + 3) = 1683.


Rupesh said: (Jul 28, 2010)  
All thing I understand but how this value of k is taken 2 can som one explain me.

Chinmaya said: (Nov 25, 2010)  
How you will get 2 ?

Sundar said: (Nov 25, 2010)  
Let (840k + 3) = 1683     [ 1683 taken from Option B]

Therefore, k = (1683-3)/840

k = 1680/840

k = 2.

Note: (840k + 3) should be divisible by 9 if we substitute k = 2.

Suchita said: (Nov 28, 2010)  
hey.....just add the digits of given options as
1+6+8+3=18 the answer is that which can be divided by 9.

remaining numbers can't divide by 9.

Priya said: (Dec 9, 2010)  
nice Suchita ......it was nice

Emu said: (Dec 25, 2010)  
suchita it"s working.........

Mehar said: (Jan 31, 2011)  
multiply all the values ==> 5*6*7*8=1680.
these are leaving 3 as a reminder so,1680+3=1683.

Uttam said: (Feb 3, 2011)  
Just try this,

Sum the digits of each option given. i.e

[A].1+6+7+7=21 [B].1+6+8+3=18


[C].2+5+2+3=12 [D].3+3+6+3=15

Now only 18 is divisible by 9

So answer should be 1683 that is option B.

Sakshi said: (Mar 2, 2011)  
nice Mehar.

Gaurav said: (Jun 19, 2011)  
You can find out whether any num is divisible by nine, 3, 4 by using divisibily test. Jus try google search for divisibility test.

Saurabh Agrawal said: (Jul 25, 2011)  
Suchita's trick is great. Very short and productive.

Shariq said: (Nov 13, 2011)  
Just divide all options by 9.

Simanta said: (Dec 22, 2011)  
Nice suchita !

M.V.Krishna/Palvoncha said: (Jan 11, 2012)  
Excellent suchita.

Gogol said: (Feb 27, 2012)  
@SUCHITA

If one of the option becomes for example..2538, then the sum of 2+5+3+8= 18 will also be divided by 9 leaving no remainder.

So, in this case there is no profit from your logic.

Anand said: (Apr 24, 2012)  
The solution of finding the option which is divisible by 9 works only in this specific case of provided options. In case, when more than 1 option is divisible by 9, the method to find out the value of k needs to be understood.

Pankaj said: (Aug 26, 2012)  
Nice Suchita.

Feroz said: (Sep 18, 2012)  
we get LCM =840 since 843 is divided by 3
when we divide it by 9 it leaves a remainder of 6
since we need a least no. divided by three which also leaves 3 remainder
we put value of k=1, 2, 3.... till we get the required no.
since

Satheesh said: (Dec 5, 2012)  
Divide each answer by 9 which one has no reminder that's answer.

Chandresh said: (Aug 7, 2013)  
But concept of finding the value of k is still not clear.

Can anyone tell the concept except logical solution?

Lali said: (Aug 23, 2013)  
It clear when we put a value of k 1 then 5,6,7,8, leave the remainder 3 and 6 in case of 9. when k=2. in this case no remainder is left.

Naveen said: (Jan 30, 2014)  
840/9 =93 and leaves reminder 3.

Now 9*93 + (3 * x +3) now find the value of x in (3* x + 3) which is divisible by 9.

If you put 2 then it is divisible by 9 i.e., (3*2+3).

So k=2.

Ramya said: (Apr 24, 2014)  
Why we are using variable k?

Rd Sharma said: (Jun 2, 2014)  
In question when number is divided by 9 no reminder. If reminder is 2 then how can solve this question. Please reply anyone.

Kranthi said: (Jun 11, 2014)  
If all the options divided by 9 then what we have to choose?

Priyanka Mishra said: (Jul 27, 2014)  
@Kranthi you can check from the options by dividing the number by 5, 6, 7, 8 and find if the required remainder is coming or not.

Kannan said: (Aug 1, 2014)  
In the below eqn.
Least value of k for which (840k + 3) is divisible by 9.

K can be K=0,1,2...

Sub K=1,
(840*1)+3 =-- Not divisible by 9.

Sub K=2,
(840*2)+3 = 187-- Divisible by 9.

Sub K=3,
(840*3)+3 =-- Not Divisible by 9.

Hence take lease no. (i.e 2).

Bujji said: (Aug 23, 2014)  
Simply divide the options with 9 and which option whose remainder is 0 it is the answer.

Lavi said: (Dec 1, 2014)  
Oh, its simple one. You have to check only options. As question said when divided by 9 leaves no remainder. Means you have to check the divisibility of 9.

Iswaryalakshmi said: (Apr 3, 2015)  
If they don't give options how can I get 'k' value? Please explain.

Anand said: (Apr 7, 2015)  
Here we have two conditions:

Conclusion 1:

The correct answer choice must be divisible by the given numbers 5, 6, 7 and 8 and leaves a remainder 3 or we can say the "correct answer choice - 3" completely divided by 5, 6, 7 and 8.

Let we look at con 1 first:

(1677 - 3)/5 1674 is not divisible by 5 it give some remainder. So simply eliminate this choice.

(1683 - 3)/5 remainder is "0",
(1683 - 3)/6 remainder is "0",
(1683 - 3)/7 remainder is "0",
(1683 - 3)/8 remainder is "0", all 4 values are satisfied the condition so it may be a correct choice.

Lets continue with next choice.

(2523 - 3)/5 remainder is "0",
(2523 - 3)/6 remainder is "0",
(2523 - 3)/7 remainder is "0",
(2523 - 3)/8 remainder is "0", all 4 values are satisfied the condition so it also may be a correct choice.

Lets continue with next choice.

(3363 - 3)/5 remainder is "0",
(3363 - 3)/6 remainder is "0",
(3363 - 3)/7 remainder is "0",
(3363 - 3)/8 remainder is "0", all 4 values are satisfied the condition so it may be a correct choice.

When we finish with the first condition we eliminate one choice and remaining we have 3 choices, to find the correct answer choice from these 3 choice we need to go for condition 2;

Conclusion 2:

The correct answer choice must be divide by 9 without any remainder.

Before go for the calculation please remind one thing, if we need to know whether the give number is completely divisible by 9 or not, simply add the digits, if it is the multiples of 9 then the given number is divisible by 9. Let we check it now.

1683 = 1 + 6 + 8 + 3 = 18 = 1 + 8 = 9 (9 is a multiple of 9 (1 x 9 = 9).

2523 = 2 + 5 + 2 + 3 = 12 = 1 + 2 = 3 (3 is not a multiple of 9). Condition false so we can eliminate this choice also.

3363 = 3 + 3 + 6 + 3 = 15 = 1 + 5 = 6 (6 is not a multiple of 9). Condition is false so we can also eliminate this option.

Finally we have only one option left that satisfies both the conditions so, that's our answer. "Choice B - 1683".

It may be look like a lengthy one but if you understand whats actually happened behind the calculations, it's simple.

We can stop our calculation Once we find choice B is correct, no need to go and option C and D, but in case we have any option like 'can not be determined' or 'insufficient data', we need to check all the options.

Hope you enjoy, got struck with any step ask me.

Good Luck. !

Mohit said: (Apr 11, 2015)  
For those who giving method of dividing options by 9 and leaving 0 as remainder. During actual tests question makers just pull out trick like they give options which are leaving 0 remainder after dividing by 9 and also these options are satisfying other conditions.

So examples like this must be solve by given method of solving equation, this is the fastest and reliable method.

Ayush said: (May 25, 2015)  
L.C.M. Of 5, 6, 7, 8 = 840.

Required number is of the form 840k + 3.

We know that it is divisible by 9.

So, 840k+3 = 9x. (x being the quotient).

Now, k is found by just putting values starting from 1, 2, 3.

If we get the lowest value which when used in k makes equation divisible by 9, it will surely be 2.

Hope it helped.

Ayush said: (May 26, 2015)  
To find value of k, see the following:

840k+3 = 0 (mod9).
30k+3 = 0 (mod9) [840/9=30 (mod9)].
30k = 6 (mod9).
3k = 6 (mod9) [30/9=3 (mod9)].

Dividing both sides by 3, we get

k = 2 (mod9).

So, the value of k = 2.

HOPE THIS ONE ALSO HELPED YOU.

Ritvik said: (Aug 8, 2015)  
Best possible way to add the digits in the option and check divisibility with 9.

Siva said: (Sep 27, 2015)  
Finding the answer with the choose options is the best method but finding the value of k has some formula. That is what should know.

Itsnoble said: (Oct 8, 2015)  
@Suchita.

What if we have 111111111? Sum equal 9 but not divisible by 9.

An exception but nice work in this case.

Mukul said: (Nov 27, 2015)  
What if there was no option for and, then how to find k?

Truptti said: (Dec 15, 2015)  
How get 840?

Vinay said: (Dec 18, 2015)  
Yes if there was no option for how to find k. I can also go with @Mukul.

Rohit said: (Dec 19, 2015)  
Please help me how we get k=2?

Vijay said: (Feb 9, 2016)  
Please tell me how you put the value k=2?

Triveni said: (Feb 25, 2016)  
You know concept of "Dividend : (Divisor*Quotient) plus Remainder".

So here 800 is divisor "k" is quotient "3" is remainder.

Manikanta said: (Jun 29, 2016)  
@Itsnoble.

111111111 is divisible by 9.

Sunita said: (Jul 8, 2016)  
If there is no option is given, and the question is asking sum of that no what will we do?

Vikram Rajput said: (Jul 23, 2016)  
Divide all options by number which is multiple in the question if remainder 0 for any option, that is your answer.

Akshay said: (Aug 21, 2016)  
Please tell me, how to find the value of k?

Sags. said: (Sep 5, 2016)  
Why are we using variable 'K'? Please tell the method not shortcut.

Tina Bhattacharjee said: (Nov 26, 2016)  
@Suchita.

18 is divisible by 6. There is no remainder. But as per the question, we should get 3 as the remainder when we divide the number by 6.

Samiul Sk said: (Dec 2, 2016)  
Find the least number which when divided by 12, leaves a remainder of , when divided by 15, leaves a remainder of 10, when divided by 16, leaves a remainder of 11.

(a)115 (b)235 (c)247 (d)475

Solve and find the answer.

Suresh Binwal said: (Dec 20, 2016)  
Any number whose digits sum is nine (9) then only the number is divisible by nine.

taking k = 1..means 843 , sum will be 8+4+3 = 15 not divisible by nine.

k = 2 is 1643 & sum of digit is 1+6+4+3 = 18 means 1+8 = 9 hence least number is k=2.

Shruti said: (Apr 7, 2017)  
@Suchita.

You are truly right this really works!

Must say this is the best Sol for this question. Thanks.

Rupam Gupta said: (Apr 27, 2017)  
Explanation of the answer is very helpful, Thank You all.

Sana said: (Jun 5, 2017)  
Very clear and effective explanation. Thanks a lot @Suchita.

Sandeep said: (Jun 7, 2017)  
Find the least number which when divided by 16,18,20,25 leaves 4 as a remainder in each case but when divided by 7 leaves no remainder.

a) 17004
b) 18000
c) 18002
d) 18004

Can anyone explain the answer?

Shiva Shankar said: (Jun 16, 2017)  
Please clarifly my doubt after find out LCM for the numbers 5, 6, 7, 8 = 840.

Required number is of the form 840k + 3.

How K = 2 comes please clarify friends.

Akshay said: (Jun 22, 2017)  
How did you get that the req number is of the form 840k+ 3?

Kiruthikabaskaran said: (Aug 8, 2017)  
Super shorcut method. Thank you so much @Suchita.

Nitesh said: (Aug 24, 2017)  
Thanks @Suchita.

Bhagirath said: (Oct 9, 2017)  
Lcm of 5,6,7, and8=840.
840÷9=93,leaves remainder =3.
multiply all the values=5*6*7*8=1680.
and leaves remainder are added =1680+3=1683.

Shubham Shah said: (Nov 11, 2017)  
Answer is just simple divide all option values by 9 which is exactly divisible is your answer. Just simple.

Munna Mudassir said: (Dec 5, 2017)  
You will not get this much time in solving the question in competitive exams. There is a shortcut in solving this.

First check divisibility of each number-3 with 5,6,7 and 8 and the divisibility of the number with 9.

a) 1677 -3=1674, it will not be divisible by 5 so no need to check with any other.
b)1683-3= 1680, Last digit is 0 hence divisible by 5. Divisible by 2 (even number) and 3 (sum = 15 is divisible by 3) hence divisible by 6.

Checking divisibility by 7-> 16-8*2= 0. hence divisible by 7.
It is divisible by 8 -> (8*21=168).
Sum of original number 1683 = 18 which is divisible by 9 hence 1683 is divisible by 9.
There is no other number smaller than 1683 Hence the answer is 1683.

Munna Mudassir said: (Dec 5, 2017)  
Another shortcut which will work in this case is checking thew divisibility by 9.
only 1683 is divisible by 9 hence the answer is 1683. In case there is more numbers divisible by you have to check it as my previous answer.

Renu said: (May 4, 2018)  
Good explanation, Thanks @Suchita.

Ram said: (May 9, 2018)  
Thanks for the explanation @Suchita.

Tg said: (Jul 1, 2018)  
2 has come because 5*6*7*8 is 1683.

Shruthi Aishwarya said: (Jul 17, 2018)  
Apply the divisibility test of 9 to the options, in this case only 1683 and 3363 are divisible by 9. Since they have asked for the least number 1683 is the answer.

Gowtham said: (Jul 25, 2018)  
Well said, thanks @Suchita.

Sudip said: (Aug 24, 2018)  
Thanks for the answer @Suchita.

Anjali said: (Aug 30, 2018)  
Check the 4 options divisibility for 9.

Option B.
1+6+8+3=18 which is divisible by 9.
While other options are not divisible by 9.

Kannan said: (Apr 20, 2019)  
How k= 2? Explain.

Prekaha Darshan said: (Apr 23, 2019)  
In simple when we go with options

Option A)1677
1677/9, remainder=3.

Option B) 1683
1683/9, remainder=0.

Option C) 2523
2523/9, remainder=3.

Option D) 3363
3363/9, remainder=3.

When 1683/5, 1683/6, 1683/7, 1683/8 leaves the remainder of 3 and 1683/9 leaves no remainder.

Hence, the correct option is B.

Shwetha said: (Jul 13, 2019)  
@Aall.

The solution is;
Check :A-1677 divisible given no 5 we get reminder 2 so this not correct.

Then next B-1683 divisible given no 5 we will get reminder 3 next.
Divisible no 6 we get reminder 3.
Divisible 7 and 8 also we get reminder 3.
Then; 1+6+8+3=18, divisible by 9 so answer is B correct.

Jitendra Kumar said: (Jul 21, 2019)  
What is the Value of k?

Shankar said: (Aug 3, 2019)  
Simply we can solve this by;

1+6+8+3=18
1+6+7+7=21
Same as other;

18 is divided by 9 and then B is correct.

Neelima said: (Sep 18, 2019)  
@All.

Hai I have a doubt, everyone is explaining from 840k+3 but here the doubt is we got lcm 840 if 5 6 7 8 divide the 840 answer is 0.

So, 840 + 3=843 now we will get remainder 3 but here 5 is divisor 840 is dividend so how you come 840k+3? Explain it.

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