LCM & HCF Quiz 1

Solution : 

Solution:

Let us first convert each length to cm. So, the lengths are 450 cm, 990 cm, and 1620 cm. Now, we need to find the length of the largest plank that can be used to measure these lengths as the largest plank will take the least time. For this, we need to take the HCF of 450, 990, and 1620.
450 = 2 x 3 x 3 x 5 x 5 = 2 x 3² x 5²
990 = 2 x 3 x 3 x 5 x 11 = 2 x 3² x 5 x 11
1620 = 2 x 2 x 3 x 3 x 3 x 3 x 5 = 2² x 3⁴ x 5
Therefore, HCF (450, 990, 1620) = 2 x 3 x 3 x 5 = 90
Thus, we need a plank of length 90 cm to measure the given lengths in the least time.

Solution : The required number leaves remainders 1 and 4 on dividing 70 and 50 respectively. This means that the number exactly divides 69 and 46.
So, we need to find the HCF of 69 (3 x 23) and 46 (2 x 23).
HCF (69, 46) = 23
Thus, 23 is the required number.

Solution : In these type of questions, we need to find the LCM of the divisors and add the common remainder (3) to it.
So, LCM (5, 7, 9, 12) = 1260
Therefore, required number = 1260 + 3 = 1263

Solution:

To find the required number, we need to find the HCF of (136-64), (238-136), and (238-64), i.e., HCF (72, 102, 174).
72 = 2³ x 3²
102 = 2 x 3 x 17
174 = 2 x 3 x 29
Therefore, HCF (72, 102, 174) = 2 x 3 = 6
hence, 6 is the required number.

Solution : The largest four digit number is 9999.
Now, LCM (15, 21, 28) = 420
On dividing 9999 by 420, we get 339 as the remainder.
Therefore, the required number is 9999-339 = 9660

Solution : LCM (6, 7, 8) = 168
So, the number is of the form 168m + 3.
Now, 168m + 3 should be divisible by 9.
We know that a number is divisible by 9 if the sum of its digits is a multiple of 9.
For m = 1, the number is 168 + 3 = 171, the sum of whose digits is 9.
Therefore, the required number is 171.

Let the common ratio be ‘m’. So, the numbers are 2m and 3m.
Now, we know that Product of numbers = Product of LCM and HCF.
=> 2m x 3m = 294
=> m² = 49
=> m = 7
Therefore, the numbers are 14 and 21.

Solution: We need to find the size of a square tile such that a number of tiles cover the field exactly, leaving no area unpaved.
For this, we find the HCF of the length and breadth of the field.
HCF (180, 105) = 15
Therefore, size of each tile = 15m x 15m
Also, the number of tiles = \frac{\text{area of field}}{\text{area of each tile}}
=> Number of tiles = \frac{(180 \times 105)} { (15 \times 15)}
=> Number of tiles = 84
Hence, we need 84 tiles, each of size 15m x 15m.

Solution : They all will whistle again at the same time after an interval that is equal to the LCM of their individual whistle blowing cycles.
So, LCM (42, 60, 78) = 2 x 3 x 7 x 10 x 13 = 5460
Therefore, they will blow the whistle again simultaneously after 5460 sec, i.e., after 1 hour 31 minutes, i.e., at 11:01:00 hours.