UPSC CDS 2 2018 Math Paper

If a number is divisible by a given set of numbers, then that number is divisible by their LCM.
16=24
36=22×32
45=32×51
48=24×31
LCM of 16, 36, 45 and 48=24×32×51=720
The required number must be divisible by 720.
The largest 4–digit number = 9999
Remainder of (

9999

720

)=639
The largest 4–digit number divisible by 720=9999−639=9360