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. !

You will not get this much time in solving the question in competitive exams. There is a shortcut in solving this.

First check divisibility of each number-3 with 5,6,7 and 8 and the divisibility of the number with 9.

a) 1677 -3=1674, it will not be divisible by 5 so no need to check with any other.
b)1683-3= 1680, Last digit is 0 hence divisible by 5. Divisible by 2 (even number) and 3 (sum = 15 is divisible by 3) hence divisible by 6.

Checking divisibility by 7-> 16-8*2= 0. hence divisible by 7.
It is divisible by 8 -> (8*21=168).
Sum of original number 1683 = 18 which is divisible by 9 hence 1683 is divisible by 9.
There is no other number smaller than 1683 Hence the answer is 1683.

Lcm of 5, 6, 7, 8 = 840.

So, divide all the numbers in the option by 840.

i) 1677/840= 840 ×1+remainder( of course gonna be more than 3
Since in question given that 5,6,7,8
when divided a number leaves the remainder 3)

Moving on to the second option:
1683/840 = 2×840(i.e 1600)+3.

(Leaves remainder 3 ---> condition satisfied)

Checking second condition ( leaves no remainder when divided by 9)
1683/9 = 187.
So, yeah the option is B) 1683.

( Also the third and fourth option leaves the remainder when divided by 9).

Firstly, find the L.C.M of 5,6,7,8. That will give us 840.
That means 840 is divisible by 5,6,7 and 8.

According to the question, the number must leave remainder 3 when divided by 5,6,7,8. So 840+3 is the number.

Now number must be divisible by 9 and not leave any remainder. But 843 doesn't divide by 9 properly so we look for another number.

840*1 + 3 no divisible by 9.
840*2 +3 = 1683. Divisible by 9 => Hence it's the answer.

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.

In simple when we go with options

Option A)1677
1677/9, remainder=3.

Option B) 1683
1683/9, remainder=0.

Option C) 2523
2523/9, remainder=3.

Option D) 3363
3363/9, remainder=3.

When 1683/5, 1683/6, 1683/7, 1683/8 leaves the remainder of 3 and 1683/9 leaves no remainder.

Hence, the correct option is B.

Take=1,2,3,4 till reminder is zero.

Let the required number be (840k+3)/9.
If k = 1 take 840(*1)+3/9=93.667=reminder is not 00
If k = 2 840(*2)+3/9=187= reminder is 00.
Take you k=2 because reminder is 0.
The solution is 840k+3=1683.
The least value of k for which (840k+3) is divisible by 9 is k=2.

@Aall.

The solution is;
Check :A-1677 divisible given no 5 we get reminder 2 so this not correct.

Then next B-1683 divisible given no 5 we will get reminder 3 next.
Divisible no 6 we get reminder 3.
Divisible 7 and 8 also we get reminder 3.
Then; 1+6+8+3=18, divisible by 9 so answer is B correct.

Let's check the options from the divisibility rule of 9. i.e; the sum of all no should be divisible by 9.

1+6+7+7=21/9 =2.222 - NOT DIVISIBLE.
1+6+8+3=18/9=2 - DIVISIBLE.
2+5+2+3=12/9=1.3 - NOT DIVISIBLE.
3+3+6+3=15/9=1.6 -NOT DIVISIBLE.

So, Answer is Option b)1683.