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

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

@Vikrant-Garg
Probability of catching the bus = Probability of getting faulty machines in the first 2 tests (this part is clear right?)
This is possible in 2 ways - if we get DD or if we get GG as the result of first 2 tests.
In your doubt, if we get DG or GD we will require to do one more test, which would make the mechanic miss the bus. So DG and GD are not favourable cases.
So total number of ways = 4!/(2! x 2!) = 6 ways.
Favourable cases = 2
Probability = 2/6 = 1/3
Is it clear ?

@sumit-agarwal
In such scenarios like finding distinct values of [x^2/n] where x can be from 1, 2, 3 ... n[1^2/n], [2^2/n] ... [(n/2)^2/n] will yield all numbers from 0 to [n/4] (means [n/4] + 1 distinct integers)Then the next set (from [(n/2 + 1)^2/n] till [n^2/n] will be all different integers (means [n/2] distinct integers)So the number of distinct integers would be [n/2] + [n/4] + 1
if n = 100,number of distinct integers would be [100/2] + [100/4] + 1 = 76
if n = 2014,number of distinct integers would be [2014/2] + [2014/4] + 1 = 1511
if n = 13number of distinct integers would be [13/2] + [13/4] + 1 = 10
Just trying to generalize a solution shared by Kamal sir (Quant Boosters - Set 1 - Q2).
You can try out with various numbers (may be smaller numbers) so that this can be verified.

42 = 2 * 3 * 7
No of zeroes in 2006! will be the highest power of 7 in 2006!
[2006/7] + [2006/49] + [2006/343] = 331
as 42 in base 42 will be 10 and will be the provider of trailing zeroes

p = prob (overnight born = boy| Next day chosen = boy)
= prob (overnight born = boy AND Next day chosen = boy) / prob (Next day chosen = boy)
= .5 * 4/T / (.5*4/T + .5 * 3/T) = 2/3.5 = 4/7 ~ .57