At first ON -> 8/3 16/3 14/3 15/2
Each word is put off after a second
So Om will flash after -> 8/3+1 = 11/3 seconds
Similarly Innovative -> 19/3
Book -> 17/3
Store -> 15/2 + 1 = 17/2
So all will together after LCM [ 11/3 , 19/3, 17/3,17/2] = 11 * 19 * 17 = 3553 SEC

x^2 - 6x + 25 = (x-3)^2 + 16
so min of sqrt (x^2-6x+25) = 4
y^2 - 8y + 25 = (y-4)^2 + 9
so min of sqrt (y^2-8y+25) will be 3
so min sum will be 7
for x=3 and y=4

|x|+2|y|+|z|=4
for |y|=0, |x|+|z|=4 so 4n=4 * 4=16 cases
for |y|=1, y=+-1 and |x|+|z|=2 so 2 * 4 * 2=16 cases
for |y|=2 y=+-2 |x|+|z|=0 so 2 * 1 = 2 cases
so 16+16+2 = 34 integer solutions.

1000 = 2^3 * 5^
When we say HCF of factors (x,y,z) = 1 ; then we mostly talk of 3 cases ..
Case -1 ; ( 1, 2^3, 5^3) ..now here ordered arrangements = 3^2*3! = 54
Case - 2 ( 1, 1, 2^3) ; (1, 1, 5^3) ; (1, 1, 2^3 * 5^3)now total arrangement of these cases is 3!/2! ( 15) = 45
case -3 when (1 , 1, 1) ; as we are taking of factors of 1000 with HCF 1 , so this has to be a case ... And it can be arranged in only n only 1 way .. so 1
So finally 54 + 45 + 1 = 100