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

1. Decompose all numbers into prime factors.
198 2
99 3
33 3
11 11
1

252 2
126 2
63 3
21 3
7 7
1

308 2
154 2
77 7
11 11
1

2. Write all numbers as the product of its prime factors
Prime factors of 198 = 2 . 32 . 11
Prime factors of 252 = 22 . 32 . 7
Prime factors of 308 = 22 . 7 . 11

3. Choose the common and uncommon prime factors with the greatest exponent
Common prime factors: 2
Common prime factors with the greatest exponent: 22
Uncommon prime factors: 3 , 11 , 7
Uncommon prime factors with the greatest exponent: 32, 111, 71

4. Calculate the Least Common Multiple or LCM
Remember, to find the LCM of several numbers you must multiply the common and uncommon prime factors with the greatest exponent of those numbers.
LCM = 22. 32. 111. 71 = 2772 1. Decompose all numbers into prime factors

198 2
99 3
33 3
11 11
1

252 2
126 2
63 3
21 3
7 7
1

308 2
154 2
77 7
11 11
1

2. Write all numbers as the product of its prime factors

Prime factors of 198 = 2 . 32 . 11
Prime factors of 252 = 22 . 32 . 7
Prime factors of 308 = 22 . 7 . 11

3. Choose the common and uncommon prime factors with the greatest exponent

Common prime factors: 2
Common prime factors with the greatest exponent: 22
Uncommon prime factors: 3 , 11 , 7
Uncommon prime factors with the greatest exponent: 32, 111, 71

4. Calculate the Least Common Multiple or LCM.
Remember, to find the LCM of several numbers you must multiply the common and uncommon prime factors with the greatest exponent of those numbers.

LCM = 22. 32. 111. 71 = 2772= 46 minutes 12 seconds.

Whenever we need to find a number which will get divisible by certain numbers completely we take lcm. On the other case, if we need to find the number which will divide certain numbers completely we take HCf.

Here in case, we need to find the number which gets completely divide by 252, 308 & 198 that is why we are calculating LCM.

Here 252,308,198.
Choose the biggest number.

308 check whether divisible by other 2 numbers i.e 252 and 198.
If not check for factors of 308 that would be lcm,
308/252 = 1.22.
308/198 = 1.55.
multiple of 308.

308*2.
Check further finally 308 * 9=2772.
2772/252 = 11.
2772/198 = 14.

Finally the answer is 2772 i.e 46min 12sec.

A complete his round in 252 seconds.
B completes his round in 308 seconds.
C completes his round in 198 seconds.

They will agian at starting together after,
LCM of 252, 308 and 198.
252 = 2 *2 *3*3*7
308 = 2 *2*7*11
198 = 2 *3*3*11
Required LCM = 2*2*3*3*7*11 = 2772 seconds = 46 minutes 12 seconds.

We take lcm because the point of time they meet, will be exactly divisible by their individual timings. 2772 is the smallest number divisible(that is when they first meet).
That is their lcm.
2772/60= 46.2 mints.
.2 mints = 12 seconds (.2*60).

So, the answer is 46 minutes 12 seconds.

Here every one should be at the same time according to question.

But after completion of 1 round not before it.

So each individual have to complete the round. That's why we have to consider their full time that's the reason to take LCM.

2 [252,308,198]
2[126,154,99]
3[63,77,99]
3[21,77,33]
7[7,77,11]
11[1,11,11]
[1,1,1]
2*2*3*3*7*11=2772

Here, 60 sec = 1 minute.
Hence,
Simply divide 2772/60.
You will get 46.
And remainder 12.
Here, the remainder is seconds.

Just deduce it like this---->>

252 ---2*126 = 2^2*63 = 2^2*3*21 = 2^2*3^2*7.
308 ---2*154 =2^2*77 = 2^2*11*7.
198 ---2*99 = 2*3*11.
Now just take the prime factors with the highest power out.
So LCM = 2^2*3^2*7*11= 2772.

The Right answer is 46 minutes and 20 seconds, this is a correct solution which I am going to explain below:

Lcm of (252,308,198) is 2772 seconds.

Converting in minutes by dividing 60,
2772/60 = 46 minutes and 20 seconds.

Actually my problem is- taking out LCM will give when they'll meet i.e. it'll give the time at which they'll meet again but it's not necessary that they meet at the starting point? I guess dean's logic is correct.