What is the least number when we divide by 5 6 7 and 8 leaves a remainder of 3 but when divided by 9 leaves no remainder?

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Given:

The least number which when divided by 5, 6, 7 and 8 leaves a remainder 3

The least number when divided by 9 remainder = 0

Concept Used:

For the least number take LCM and also check that all conditions are satisfying or not

For same remainder in each case add the remainder in the LCM 

Calculation:

First we have to find the LCM of 5, 6, 7, 8

5 = 51

6 = 21 × 31 

7 = 71 

8 = 23 

LCM of 5, 6, 7, 8 is 23 × 31 × 51 × 71 = 840

For same remainder we have to add 3 to the LCM

⇒ 840 + 3 

Now check weather it is divisible by 9 or not

843/9 = Not divisible by 9

Now we have to take next multiple of LCM and then add 3

840 × 2 + 3

⇒ 1683

Divide it by 9

1683/9 = 187

⇒ 1683 is divisible by 9 

⇒ 1683 is the least number when divided by 5, 6, 7, 8 gives remainder 3

The least number must be divisible by 9 

For divisibility of 9 sum of digits of the number must be divisible by 9 

Checking it by options:

Option 1) 1677

1 + 6 + 7 + 7 = 21

⇒ 1677 is not divisible by 9

Option 2) 1683

1 + 6 + 8 + 3 = 18

⇒ 1683 is divisible by 9

Option 3) 2523

2 + 5 + 2 + 3 = 12

⇒ 2523 is not divisible by 9

Option 4) 3363

3 + 3 + 6 + 3 = 15

⇒ 3363 is not divisible by 9

∴ The least number is 1683 which when divided by 5, 6, 7 and 8 leaves a remainder 3, but when divided by 9 leaves no remainder.