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

There should be given in the question that one factor is 14 and 13 is the another factor number.

Because, If the no. are x and y,
We can take two conditions according to question;

Condition 1:

x = 23 * 14.
y = 23 * 13 .

Condition 2:

x = 23 * 1
y = 23 * 14 * 1.

Then the only lcm will be 23 * 14 * 13, and then only you can write in the question that 14 and 13 are other two factors.

According to this question, the answer can be;

1)
x = 23*14.
y = 23*13.
So in this 23 * 14 will be greater.

2)
x = 23.
y = 23 * 13 * 14.

So, 23 * 13 * 14 will be greater.

According to me, the question is not complete and to complete it read my first line.

Hope you understand my explanation, Thankyou.

Let us say, the two numbers are: 23 x a and 23 x b.

Why? because 23 is the highest common " factor " of the two numbers. If you still don't understand why Study the concept of factors properly.

Now we know that,

"H.C.F is one of the factors of L.C.M". (That's why in the question they have mentioned, " the 'other two' factors are 13 and 14. That means the three factors of LCM are 23, 14, 13)" .

Therefore LCM = 23 x 14 x 13.

Now we know that, " Product of two numbers = Product of their H.C.F. and L.C.M ".
Therefore, (23 x a) x (23 x b) = 23 x 23 x 14 x 13.
Therefore, a x b = 14 x 13.
Therefore, a x b = 2 x 7 x 13.

Now there are more than one values possible for a and b which satisfy the above equation. But, only if we use a = 14 and b = 13 for the two numbers (23xa and 23xb), we will get the HCF = 23. For all other values of a and b, we won't get 23 as the HCF. Try it!

Therefore, The two numbers are 23 x 14 and 23 x 13.

Hope it is clear now!

I understood it perfectly.

Here 23 is HCF. So it will also be a factor of LCM. And other two factors are given as 13 and 14 so now lcm will be 13 * 14 * 23. But we want to find numbers. So 1st number will be 23 * 13 because 23 is HCF and 13 is given as the 2nd factor of LCM. And the second number will be 23 * 14.

I hope you guys will understand this.

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.

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.

@Chiya.

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

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.

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.

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.

I didn't understand this method.

Please, can anyone explain me in an understanding manner?