There are five hobby clubs in a college viz. photography, yachting, chess, electronics and gardening. The gardening group meets every second day, the electronics group meets every third day, the chess group meets every fourth day, the yachting group meets every fifth day and the photography group meets every sixth day. How many times do all the five groups meets on the same day within 180 days?

Given, length of four metal roads are 78, 104, 117 and 169 cm.
Now,
78 = 13 x 2 x 3,
104 = 13 x 2 x 2 x 2,
117 = 13 x 3 x 3,
169 = 13 x 13

Length of each piece of rod as long as possible, HCF = 13 CM
? Number of piece = 6 + 8 + 9 + 13 = 36

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.

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.

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.

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

The time at which all the three persons meet will be the LCM of the time taken by each person individually to complete one round.

252 = 2² x 7¹ x 9¹
308 = 2² x 7¹ x 11¹
198 = 2¹ x 9¹ x 11¹

? LCM of 252, 308 and 198 = 2² x 7¹ x 9¹ x 11¹ = 2772
So, A, B and C will again meet at the starting point in 2772s i.e., 46 min and 12s

x² - y² - z² - 2yz = x² - (y + z)² = (x+ y + z)(x - y - z)
x² - y² + z² + 2yz = (x + z)² - y² = (x + y + z) (x + y - z)
x² + y² - z² - 2xy = (x - y)² - z² = (x - y + z) (x - y - z)
? LCM = (x + y + z) (x - y - z) (x - y + z)

The pieces cut from four rods is least, so length of the pieces is the HCF of 52, 65, 78 and 91 which is 13.
? Number of pieces cut =(52 + 65 + 78 + 91)/13 = 22

The time which all the three persons meet will be the LCM of the time by each person individually to complete one round.
252 = 2² x 7¹ x 9¹
308 = 2² x 7¹ x 11¹
198 = 2¹ x 9¹ x 11¹
? LCM of 252, 308 and 198 = 2² x 7¹ x 9¹ x 11¹ = 2772
So,A,B and C will again meet at the starting point in 2772 i.e.,46 min and 12s.

HCF of 403,456 and 496 = 31
Now, the number of bottles required to put 403 L of petrol = 403/31 = 13
Similarly,the number of bottles required to put 456 L of diesel = 465/31 = 15
The number of bottles required to put 496 L of mobile oil = 496/31 = 16
? Least number of bottles required having 31 L as capacity = 13+15 + 16 = 44