NCERT Class 6 Mathematics for blind and Visually Impaired Students.

We come across a number of situations in which we make use of the concepts of HCF and LCM. We explain these situations through a few examples.

Example 12

Two tankers contain 850 litres and 680 litres of kerosene oil respectively. Find the maximum capacity of a container which can measure the kerosene oil of both the tankers when used an exact number of times.

Solution :
The required container has to measure both the tankers in a way that the count is an exact number of times. So its capacity must be an exact divisor of the capacities of both the tankers.
Moreover, this capacity should be maximum. Thus, the maximum capacity of such a container will be the HCF of 850 and 680.

It is found as follows by prime factorization:

Hence,
850 = 2 × 5 × 5 × 17
= 2 × 5 × 17 × 5

and
680 = 2 × 2 × 2 × 5 × 17
= 2 × 5 × 17 × 2 × 2

The common factors of 850 and 680 are 2, 5 and 17.
Thus, the HCF of 850 and 680 is 2 × 5 × 17 = 170.
Therefore, maximum capacity of the required container is 170 litres.
It will fill the first container in 5 and the second in 4 refills.

Example 13

In a morning walk, three persons step off together. Their steps measure 80 cm, 85 cm and 90 cm respectively. What is the minimum distance each should walk so that all can cover the same distance in complete steps?

Solution :
The distance covered by each one of them is required to be the same as well as minimum. The required minimum distance each should walk would be the lowest common multiple of the measures of their steps. Can you describe why? Thus, we find the LCM of 80, 85 and 90. The LCM of 80, 85 and 90 is 12240.

The required minimum distance is 12240 cm.

Example 14

Find the least number which when divided by 12, 16, 24 and 36 leaves a remainder 7 in each case.

Solution :
We first find the LCM of 12, 16, 24 and 36 as follows :

Thus, LCM = 2 × 2 × 2 × 2 × 3 × 3 = 144

144 is the least number which when divided by the given numbers will leave remainder 0 in each case. But we need the least number that leaves remainder 7 in each case.

Therefore, the required number is 7 more than 144. The required least number = 144 + 7 = 151.

EXERCISE 3.7

Q1. Renu purchases two bags of fertiliser of weights 75 kg and 69 kg. Find the maximum value of weight which can measure the weight of the fertiliser exact number of times.

A1. 3 kg

Q2. Three boys step off together from the same spot. Their steps measure 63 cm, 70 cm and 77 cm respectively. What is the minimum distance each should cover so that all can cover the distance in complete steps?

A2. 6930 cm

Q3. The length, breadth and height of a room are 825 cm, 675 cm and 450 cm respectively.
Find the longest tape which can measure the three dimensions of the room exactly.

A3. 75 cm

Q4. Determine the smallest 3-digit number which is exactly divisible by 6, 8 and 12.

A4. 120

Q5. Determine the greatest 3-digit number exactly divisible by 8, 10 and 12.

A5. 960

Q6. The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they change simultaneously at 7 a.m., at what time will they change simultaneously again?

A6. 7 minutes 12 seconds past 7 a.m.

Q7. Three tankers contain 403 litres, 434 litres and 465 litres of diesel respectively. Find the maximum capacity of a container that can measure the diesel of the three containers exact number of times.

A7. 31 litres

Q8. Find the least number which when divided by 6, 15 and 18 leave remainder 5 in each case.

A8. 95

Q9. Find the smallest 4-digit number which is divisible by 18, 24 and 32.

A9. 1152

Q10. Find the LCM of the following numbers :
(a) 9 and 4
(b) 12 and 5
(c) 6 and 5
(d) 15 and 4

Observe a common property in the obtained LCMs. Is LCM the product of two numbers in each case?

A10.
(a) 36
(b) 60
(c) 30 (d) 60

Here, in each case LCM is a multiple of 3
Yes, in each case LCM = the product of two numbers

Q11. Find the LCM of the following numbers in which one number is the factor of the other.
(a) 5, 20
(b) 6, 18
(c) 12, 48
(d) 9, 45
What do you observe in the results obtained?

A11.
(a) 20
(b) 18
(c) 48
(d) 45
The LCM of the given numbers in each case is the larger of the two numbers.