Three pieces of timber 42 m, 49 m and 63 m long have to be divided into planks of the same length, What is the greatest possible length of each plank?

Biggest measure = H.C.F. of 403, 434 and 465
= 31 kg

L.C.M of 6, 7, 8, 9, 12 is 504
So, the bells will toll together after 504 sec.
In hour, they will toll together
= (60 x 60) / 504 times
= 7 times

H.C.F. of 21, 42, 56 = 7
Number of rows of mango trees, apple trees and oranges trees are 21/7 = 3, 42/7 = 6 and 56/7 = 8
? Required number of rows = (3 + 6 + 8 ) = 17

Let the number be 81a and 81b where a and b are two numbers prime to each other.
? 81a + 81b = 1215
? a + b = 1215/81 = 15

Now, find two numbers, whose sum is 15, the possible pairs are (14, 1) , (13, 2), (12, 3), (11, 4), (10, 5), (9, 6 ), (8, 7) of these the only pairs of numbers that are prime to each other are (14, 1) , (13, 2), (11, 4), and (8, 7).

Hence, the required number are
(14 x 81, 1 x 81), (13 x 81, 2 x 81), (11 x 81, 4 x 81), (8 x 81, 7 x 81)
or (1134, 81), (1053, 162), (891, 324), (648, 567)
So, there are four such pairs .

Let the number be 16a and 16b, where a and be are two number prime to each other.

? 16a x 16b = 7168
? ab = 28
Now, the pairs of number whose products is 28, are (28, 1) : (14, 2), (7,4)
14 and 2 which are not prime to each other should be rejected.
Hence, the required answer
= (448 + 16) + (112 + 64) = 640

L . C . M of 2, 4, 6, 8, 10 and 12 is 120
So the bells will toll together after 120 second i,e 2 minutes. In 30 min. they will together in (30/2) + 1 times, i.e 16 times.

Let the least multiple of 7 be x, which when divided by 90 leaves the remainder 4. Then, x is of the form 90k + 4
Now, the minimum value of k for which 90k + 4 divisible by 7 is 4
? x = 90 x 4 + 4
= 364

In any 4 consecutive natural numbers:

Two numbers will be divisible by 2, one of which will be divisible by 4, at least one number will be divisible by 3.

These facts give us a lower bound of 2 x 4 x 3 = 24 for the solution.

But 24 is also the product of the first 4 consecutive natural numbers, so it is also an upper bound.

Since 2, 3, 5, 17 are prime numbers and
The given expression = (2³)²⁰ x (3 x 5)²⁴ x (17)¹⁵
= 2⁶⁰ x 3²⁴ x 5²⁴ x 17¹⁵

So the total number of prime factors in the given expression is (60 + 24 + 24 + 15 ) = 123.

Interval of changes = (L . C . M of 48, 72, 108) sec. = 432 sec.
So, the lights will simultaneously changes after every 432 seconds i,e 7 min. 12 sec.
So, the next simultaneously changes will take place at 8 : 27 : 12 hrs.