Question

Find the greatest number of 4 digit which is divisible by both 8 and 11

Answer 1

Method 1:
Find LCM of 8 and 11
As both are coprime numbers
LCM (8,11)=8×11=88
Therefore number will be multiple of 88.
Largest 4 digit number is 9999 and is not divisible by 8 and 11
Therefore expected number will be less than 9999
100th multiple of 88=88×100=8800.
110th multiple of 88=88×110=9680
111th multiple of 88=88×111=9768
113th multiple of 88=88×113=9944
114th multiple of 88=88×114=10032
Above all 5 numbers are divisible by 8 and 11
But largest 4 digit number from above 5 is 9944
Answer is 9944
Method 2:
Divisibility rule of 8= Last three digits of number should be divisible by 8
Divisibility rule of 11=Difference between sum of odd digits and even digits of number should be 0 or divisible by 11.
Now, largest 4 digit number is 9999
If 9999 divided by 8 gives remainder 7
Therefore largest 4 digit number divisible by 8 is 9999–7=9992
But it is not divisible by 11
Now we will check numbers less than 9992 which are divisible by 11
They are 9988,9977,9966,9955,9944,9933..
Among above numbers we have to check which number is divisible by 8
Only 9994 is divisible by 8 as it's last 3 digits are divisible by 8.
Therefore the number is 9944