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

@Ramesh.

We know that all the factors of the number A are also the factors of all the multiples of the number A.
In the given problem, 13 and 14 are factors of the Least Common Multiple of 2 numbers, therefore 13 and 14 are also considered as factors of the two numbers.
The HCF of 2 numbers = 23.
Therefore the required numbers are 23 x 13 = 299 and 23 x 14 = 322.
Therefore 23x14=322 is the highest number.
HCF of 299 and 322 = 23,
LCM of 299 and 322 = 4186.
13 and 14 are the factors of 4186.

Statement 1:- We know that if 2 numbers are co-primes, then their LCM is equal to their product.
Statement 2:- We also know that if 2 numbers are co-primes,then their HCF is 1.

In the given question, 13 and 14 are the factors of the LCM of 2 numbers. Therefore it is understood that the 2 numbers are the multiples of 13 and 14.

23 is the HCF of 2 numbers. Therefore 2 numbers are not co-primes.
Given that 23 is the common factor of the 2 numbers - 23x13 = 299 and 23 x 14 = 322.
LCM of 299 and 322 = 4186.13 and 14 are the factors of 4186.
HCF of 299 and 322 = 23.
Therefore, 23 x 14 = 322 is the highest number.

Not getting this. Please can anybody explain me clearly?

I didn't understand this method.

Please, can anyone explain me in an understanding manner?

It's very simple. Without any calculation, it can easily be solved. Look at numbers.

13 & 14, the last digit of them are 3 & 4. All numbers. Except 322 are not divisible by 14. Hence 322 is the required number.

Let Nos: x, y.
HCF = 23.

Other two factors of their L.C.M are 13 and 14 => x13 = LCM, y14 = LCM.
We know HCF *LCM = Product of Nos.
23 * x13 = xy = 23 * y14.
Take 1st pair, 23 * x13 = xy which gives y = 23 * 13.
2nd pair xy = 23 * 14.
x = 23 * 14 = 322 Highest.

As per the given information of the question,
We know HCM is 23 and LCM is 23 * 13*14.
Hence No 1 = 23 * 13, No2 = 23 *14.
Hence, 23*14 is the largest Number.

@Chiya.

Good explanation and easy to understand, thanks for your explanation.

HCF is 23.

LCM is 13 * 14 * 23 (as it is given 13 & 14 are OTHER FACTORS),

Both the numbers have 13 & 14 as their factors, Hence the greater number should be divisible by 14 & the smaller number should be divisible by 13.

The answer is 322 (divisible by 14) as 322 = 23 * 14.

a * b = lcm * (hfc)^2.
a * b = lcm * hfc * hfc.
a * b = (14 * 13) * 23 * 23,
a * b = (14 * 23) * (13 * 23),
a = 14*23 and b = 13 * 23,
a > b,

=>a = 322.

Correct me if I am wrong.