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

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. !

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.

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.

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.

Best possible way to add the digits in the option and check divisibility with 9.

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.

@Suchita.

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

An exception but nice work in this case.

What if there was no option for and, then how to find k?

How get 840?

Yes if there was no option for how to find k. I can also go with @Mukul.