Find the least number which when divided by 3 5 8 and 12 leave 2 as remainder in each case

The least number which when divided by 5, 6 , 7 and 8 leaves a remainder 3, but when divided by 9 leaves no remainder, is:

Answer: Option B

Explanation:

L.C.M. of 5, 6, 7, 8 = 840.

Least value of k for which (840k + 3) is divisible by 9 is k = 2.

Therefore, k = (1683-3)/840

k = 1680/840

k = 2.

Note: (840k + 3) should be divisible by 9 if we substitute k = 2.

remaining numbers can't divide by 9.

Sum the digits of each option given. i.e

[A].1+6+7+7=21 [B].1+6+8+3=18

[C].2+5+2+3=12 [D].3+3+6+3=15

Now only 18 is divisible by 9

So answer should be 1683 that is option B.

If one of the option becomes for example..2538, then the sum of 2+5+3+8= 18 will also be divided by 9 leaving no remainder.

So, in this case there is no profit from your logic.

Can anyone tell the concept except logical solution?

Now 9*93 + (3 * x +3) now find the value of x in (3* x + 3) which is divisible by 9.

If you put 2 then it is divisible by 9 i.e., (3*2+3).

So k=2.

K can be K=0,1,2...

Sub K=1,
(840*1)+3 =-- Not divisible by 9.

Sub K=2,
(840*2)+3 = 187-- Divisible by 9.

Sub K=3,
(840*3)+3 =-- Not Divisible by 9.

Hence take lease no. (i.e 2).

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. !

So examples like this must be solve by given method of solving equation, this is the fastest and reliable method.

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.

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.

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

An exception but nice work in this case.

So here 800 is divisor "k" is quotient "3" is remainder.

111111111 is divisible by 9.

18 is divisible by 6. There is no remainder. But as per the question, we should get 3 as the remainder when we divide the number by 6.

(a)115 (b)235 (c)247 (d)475

Solve and find the answer.

taking k = 1..means 843 , sum will be 8+4+3 = 15 not divisible by nine.

k = 2 is 1643 & sum of digit is 1+6+4+3 = 18 means 1+8 = 9 hence least number is k=2.

You are truly right this really works!

Must say this is the best Sol for this question. Thanks.

a) 17004
b) 18000
c) 18002
d) 18004

Can anyone explain the answer?

Required number is of the form 840k + 3.

How K = 2 comes please clarify friends.

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.

Option B.
1+6+8+3=18 which is divisible by 9.
While other options are not divisible by 9.

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.

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.

1+6+8+3=18
1+6+7+7=21
Same as other;

18 is divided by 9 and then B is correct.

Hai I have a doubt, everyone is explaining from 840k+3 but here the doubt is we got lcm 840 if 5 6 7 8 divide the 840 answer is 0.

So, 840 + 3=843 now we will get remainder 3 but here 5 is divisor 840 is dividend so how you come 840k+3? Explain it.

Check the options whether it is divisible by 9.

Only 1683 is divisible by 9.

And then after 1680+3 = 1683
The answer is 1683.

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.

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.

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.