Free CAT Questions

You get official solutions with each question, and our insights to help you track your preparation.

Click here!
RELATED TAGS
ICICI Amity RBI Great Lakes CAT Verbal FMS Delhi NMAT General Knowledge CAT Quant

CAT

Discuss your CAT preparation strategies here. Find BSchools which accept CAT results. Ask queries and get answers from current students and alumni.

The sum sides of a triangle is 27 m .

Find the number of scalene triangles.


  • 7Comments
  • @kvaish89 How do you get total number of triangles?. 2m.
  • @Bulbuul for even perimeter n, number of triangles = ceil.... 59s.
  7 Comments
SourabhSur123
Sourabh Sur @SourabhSur123 26

@Bulbuul                                                                                                                       I kinda counted all the possible triangles, just made a bit of adjustments here and there. Now in scalene traingle all the sides should be unequal and of course the criterion for being a triangle a+b > c should be fulfilled. a + b + c = 27. As you can see none of the sides can be greater than 13 then the criterion a + b > c will fail. So start checking from 1 to 13 i.e. keeping 1 as one of the sides and varying other two sides according to the below 3 criteria- 1. a + b + c = 27 2. a+b > c 3. a , b, c none of them being equal. There will be some common cases which you need to eliminate by seeing, and it won't take much time to get all the 12 triangles. If anyone has a better solution then please kindly share.                                                                               PS : I assumed that the question asks for only integer sides.


kvaish89
Kshitiz Vaish @kvaish89 23

total number of triangles = ceiling((27+3)^2/48) = 19

number of cases when any two sides are equal = integral part of ((27+1)/4) = 7

required = 19-7=12

Bulbuul
Bulbuul Dev @Bulbuul 215

@SourabhSur123 Its like 

Toptal number of triangles - ( equilateral+ isoscale) = 12

kvaish89
Kshitiz Vaish @kvaish89 23

@Bulbuul for even perimeter n, number of triangles = ceiling(n^2/48)

for odd perimeter n, number of triangles = ceiling((n+3)^2/48)

so (27+3)6/48 = 18.75 = 19

2m ago 2.5k CAT Verbal

Official verbal ability thread for CAT 2014

Amidst IIFT,NMAT and other exams preparation we have come so far and near to CAT 2014. Not so many Days (Literally) f...

Scarcely had he entered the room than the mob attacked him.

Sentence Correction

  • 3Comments
  • case of co-relative conjunction !! the correct pairing is.... 4h.
  • throw some light on it....... 2m.
rafaelnadal
Indranil Roy @rafaelnadal 581

case of co-relative conjunction !! the correct pairing is scarcely-when , so 'than' should be replaced by 'when' !!

The admiral ____ his order to attack when he saw the white flag raised by the enemy sailors; he was relieved that he could bring an end to the ____ .  (A) reiterated - hostilities  (B) countermanded - fighting  (C) commandeered - truce  (D) renounced - hiatus  (E) confirmed - aggression 

10 people answered this question.
1 Comments
You can view the comments after answering or skipping this question.

33m ago 1.5k FMS Delhi

[Official] [2014-16] FMS Delhi: Aspirants Admissions Discussion

Hi All, Welcome to the official discussion forum of FMS Delhi 2014-16 discussions. The Red Building of dreams as we a...

I was reading about one of the posts on "All I wanted to speak about CAT". Normally this post always inspires, but I read something dreadful today. The post was about a guy from FMS and I am a 60 percentiler him. We both have 70 % in 12th and 60% in Grad. This guy said that he was finding it difficult to find an on campus job because of his bad acads. 

I request you guys to shed some light on this matter. Is it really over for 60 percentilers like me. Is it like even if we slog our asses and somehow make it to a reputed B-School, struggle won't be over for us? This guy had written that the companies did not short list him due to bad grades and it was only after 40 companies that one of them short listed him.

I want to know if this is what really happens in a top B-school and even if you perform well there and even if you have done a lot of extra - curricular activities companies might not short list you? I feel very agitated due to this. I request you guys to please share whatever info you have on this.

  • 14Comments
  • And thanks btw.. Already got placed :grin:. 2h.
  • FMS Delhi.... can't reveal till Mid- November. :grin:. 33m.

bull  Mock 6

Qa 46/43 126 ..

Va 28/18 44 ..

OA 74/61 170

All the hard work done in Qa is being wasted in va.. hell man..


  • 16Comments
  • my stategy is simple 51------>100 in 1.5hrs, score utna b.... 34m.
  16 Comments
neerajyadav.cat
Neeraj Yadav @neerajyadav.cat 10

my stategy is simple 51------>100 in 1.5hrs, score utna bhi nahi hai par sudhar raha hai dheere dheere.

MC1502-N

QA:30A 28C :82 VA :15 A 13C : 36 OA :118 :/ Has anybody else given?



  • Was it tough?. 38m.
  • At par with the latest aimcats I think. There were just n.... 37m.

37m ago 210 CAT Quant

[Official] LR-DI Thread for CAT 2014

Hi, Puys CAT 2013 is finally over. Lets start the Preparation for CAT 2014. :grin: Please use this thread for all kin...

4 friends P,Q,R and S are seated on 4 seats on the edge of a circular merry go round, equidistant from each other facing the centre of the merry go round .Initially P is facing the South , R is facing the West , S is facing the East and Q is facing the North.The merry go round rotates in the clockwise direction rotating 45 ° in Step 1, 90 ° in Step 2, 135 ° in Step 3, 180 ° in Step 4, 45 ° in Step 5, 90 ° in Step 6... and the sequence of steps continues in this way If Q and R were to interchange their sitting positions before Step 1,then at the end of Step 45 , who would be seated to the immediate right of the person seated in the South East Direction ?

  • 3Comments
  • Q?. 3h.
  • Q. 37m.

Can anyone explain me the percentile di set? I cannot understand the solution

1h ago 162

The PaGaLGuY UnderDogs Team 2014

Do you love taking up the challenge where you are considered as a person who cannot achieve the pinnacle easily but t...

A and B run a circular track and start from same point O and run in same direction . The ratio of speed of A to B is 6:1  .  Find number of times A is ahead of B by a quarter of length of track till they meet for the first time at O  ?  

Pls help guys !

  • 1 Like   23Comments
  • Bhai Forest Gump hi samjh lia aapne toh dono ko. Infinite.... 1h.
  • haha, nahi yaar, was just confused coz nothing is mention.... 1h.
  23 Comments
ashishsingh_courage
Be Phenomenal OR Be Forgotten @ashishsingh_courage 258

Bhailog, sorry for asking a lame doubt, but we are considering the length of the track here as 6X-1X = 5X na? (only then A will meet B after running 6X distance and B running 1X distance and they meet at O 5th time). But what if the radius of the circular track was infinite?   As there is nothing given about it, I am just confused!   

aashu09
Aashish @aashu09 122

Bhai Forest Gump hi samjh lia aapne toh dono ko. Infinite radius?

ashishsingh_courage
Be Phenomenal OR Be Forgotten @ashishsingh_courage 258

haha, nahi yaar, was just confused coz nothing is mentioned abt length of track  

1h ago 234

MBA Prep articles

With about a month or so to go, the question that junta is asking at this point is not "Do I have it in me to crack C...


Highest power of a number in a factorial


Questions on the highest power of a number in a factorial are some of the easiest questions one can hope to get from Number Systems. Even though not many questions are asked on this topic, the few that are asked can be solved easily by understanding the concept. Missing out on these questions is unpardonable as far as CAT is concerned.

Consider the following example:

What is the highest power of 2 in 80! ?

To solve this, we have to first understand how 80! Is constructed.

80! = 1*2*3*4*....80. From this, we can see that there are 80/2 = 40 multiples of 2 in 80!. Each of these multiples contains one power of 2. Similarly, there are 80/4 = 20 multiples of 4 in 80!. Since  4=2^2 , each of these 20 numbers contains an additional power of 2. Similarly, there are 80/8 = 10 multiples of 8 in 80!. Since  8=2^3 , each of these 10 multiples contains another additional power of 2. Similarly, there are 80/16 = 5 multiples of 16 in 80!. Since  16=2^4 , each of these 5 numbers contains an additional power of 2. Similarly, there are [80/32] = 2 multiples of 32 in 80!. Since  32=2^5 , each of these 2 numbers contains an additional power of 2. Similarly, there is [80/64] = 1 multiple of 64 in 80!. Since  64=2^6 , this number contains an additional power of 2.

Counting all the numbers, we can see that we have a total of 40+20+10+5+2+1 = 78 powers of 2 in 80!.

In general,

Formula: 

The highest power of a prime number p in x! is given by the expression [x/p]+[x/p^2]+[x/p^3]+[x/p^4]+... where [x] is the greatest integer less than or equal to x.

Now, consider the following example:

What is the highest power of 8 in 80! ?

If we solve the problem using the method used for solving case 1 ( highest power of 2 in 80! ), we get the highest power of 8 = [80/8] + [80/64] = 10+1 = 11.

This is wrong because we are missing out on some powers of 2 and 4, which when combined give a power of 8.

To solve this, we have to realize that  8=2^3 . Now, if we solve for the highest power of 2 in 80! and club every 3 powers, we get the number of powers of 8.

Since the highest power of 2 in 80! = 78, we get the highest power of 8 in 80! = [78/3] = 26.

Point to be noted here is that the method used in the first case works if the number whose highest power is being asked is a prime number. For higher powers of the number, it has to be prime factorized first. To make the point clearer:

In general,

Formula: 

The highest power of a prime number p raised to n in x! is given by the expression [{[x/p]+[x/p^2]+[x/p^3]+[x/p^4]+....}/n].

Consider another example:

What is the highest power of 10 in 70! ?

Since 10 is not prime, first prime factorize 10.

10 = 2*5.

Highest power of 2 in 70! = [70/2] + [70/4] + [70/8] + [70/16] + [70/32] + [70/64] = 35+17+8+4+2+1 = 67.

Highest power of 5 in 70! = [70/5] + [70/25] = 14+2 = 16.

Now, combining one power of 2 and one power of 5 gives one power of 10. Therefore, combining 16 powers of 5 and 16 powers of 2 gives 16 powers of 10. We still have some powers of 2, but we do not have any more powers of 5. Therefore, we cannot obtain any more powers of 10. The highest power of 10 in 70! is therefore, 16.

In general, there are always less number of powers of the highest prime number. Therefore, a short-cut can be to calculate only the highest power of the largest prime factor.

The takeaway from this example is:

Formula: To find the highest power of a composite number in a factorial, prime factorize the number and find the highest powers of each of the prime factors. The highest power of the composite number is the minimum value among the powers of each of the prime factors.

 

Some more solved examples:

1) What is the highest power of 5 in 125!?

Solution: [125/5] + [125/25] + [125/125] = 25+5+1 = 31.

2) What is the number of zeroes at the end of 150!?

Solution: The number of zeroes at the end of a number is nothing but the highest power of 10 in the number.

Highest power of 10 in 150! = min(Highest power of 5 in 150! , Highest power of 2 in 150!).

Answer = 37.

3) What is the highest power of 8 in 25!?

Solution: 8 =  2^3 . The highest power of 2 in 25! = [25/2]+[25/4]+[25/8]+[25/16] = 12+6+3+1=22. Therefore, the highest power of 8 = [22/3] = 7.

Maruti is an alumnus of IIT Bombay and IIM Ahmedabad and is the co-founder of http://cracku.in

  • 1 Like   3Comments
  • Yeah. We will discuss some such problems in the days goin.... 2d.
  • very helping.. i was doing it wrong by not spliting the 1.... 1h.
harshcat91
Harsh Vardhan Singh @harshcat91 2.4 k

Great work. As you have discussed this concept try to post the ones for no of zeros skipped also in factorial. It seems lot of people find it new but its a application of the same.

cracku.in
Cracku @cracku.in 44

Yeah. We will discuss some such problems in the days going forward

humanerror
Shreyans Shrivastava @humanerror 1

very helping.. i was doing it wrong by not spliting the 10 and getting the ans wrong!! thanks kudos

1h ago 1.1k

[2012-2013] CAT Coaching Classes / Test Series Queries

Hi Friends, Carrying forth the tradition , here we come up with a new thread for queries related to coaching classes ...

guys which provide best coaching in trichy?

1h ago 44

Executive Post Graduate Diploma in International Business from IIFT

All, Does anyone have any idea on Executive Post Graduate Diploma in International Business program from IIFT ? EPGDI...

Did anyone got call/email/letter from IIFT regarding the results for executive MBA?

  • 16Comments
  • Going to join frm 2mrrw.. C u guys der... 1h.
  16 Comments

Can anyone explain Q15 of QADI?

  • there is no number which has 5^30 in its factorial. see.... 2h.
  • Thanks a lot @nik hil. 1h.
managerbhai
nik hil @managerbhai 1.5 k

there is no number which has 5^30 in its factorial. 

see this for yourself.

100 has 5^24

105 will have 5^25

110 -> 5^26

115->5^27

120->5^28

125->5^31

hence 30 is the answer.