LCM of 12, 15, 20 and 35 = 420
The greatest 4 digit number = 9999
Now, divide 9999 by 420, we get 339 as remainder
Hence, Required number = 9999 - 339 = 9660

LCM of 12, 18, 21 and 28 = 252
The greatest 4 digit number = 9999
Now, divide 9999 by 252, we get 117 as remainder
Hence, Required number = (9999 - 117) + 3 = 9931

LCM of 12, 15, 20 and 35 = 420
The 4-digit smallest number = 1000
Now, divide 1000 by 420, we get 160 as remainder
Hence, Required number = 1000 +(420 - 160) = 1260

LCM of 12, 18, 21 and 28 = 252
The 4-digit smallest number = 1000
Now, divide 1000 by 252, we get 244 as remainder
Hence, Required number = 1000 +(252 - 244) + 3 = 1011

The LCM of 5,6,8,9,12 = 360
The required number = 360Z+1 (Z is a positive number)
Divide 360, by 13, we get 27 (Quotient) and 9 (Remainder)
∴ 360Z+1 = (13 x 27 + 9)Z + 1
360Z + 1 = (13 x 27 x Z) + (9Z + 1)
Now, this number has to be divisible by 13. Whatever may be the value of Z the portion (13 x 27Z) is dalways divisible by 13. Hence we must choose that least value of Z which will make (9Z +1) divisible by 13. Putting Z equal to 1,2,4,5 etc in succession, we find that Z must be 10
∴ the required number = 360 x Z + 1 = 360 x 10 + 1 = 3601

Shortcut:
The product of 'n' numbers = (HCF of each pair) ^n-1 x (LCM of n numbers)
Using the shortcut, we get
The product of 4 numbers = (3)^4-1 x 116
∴ The product of 4 numbers = (3)³ x 116 = 27 x 116 = 3132