HCF & LCM

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GETTING STARTED

HCF:

☛ HCF stands for Highest Common Factor. As the name suggests, it is the largest number which divides all of the given numbers.

☛ Let us take two numbers, 12 and 18.
2, 3, 6 are the numbers which divide them both (common factors). Hence the HCF of the two numbers is 6.

LCM:
☛ LCM stands for Least Common Multiple. As the name suggests, it is the smallest number which is a multiple of all the given numbers.


☛ Let us take two numbers, 12 and 18.
36, 72, 108 are a few numbers which are multiples of both 12 and 18. But since 36 is the least of all, it is the LCM of given numbers.

Method to find the HCF for given numbers:
Prime factorization Method:
☛ Break down all the numbers into their corresponding prime factors raised to their powers
☛ Select the distinct common prime factor with least power and multiply all of them.

Eg. 2160, 3420
2160 = 8 x 27 x 10

= 23 x 33 x 21 x 51  

= 24 x 33 x 51

 

3420 = 2 x 171 x 10 = 2 x 9 x 19 x 10

=   21 x 32 x 191 x 21 x  51

=  22 x 32 x 51 x 191

Note: In the above process of splitting numbers into their prime factors, we started off with numbers which are their factors for sure and then made them into smaller numbers further finding their other factors.

The common factor is: 22 x 32 x 51 
Therefore the HCF = 4 x 9 x 5 = 180

Method to find the LCM for given numbers:
Prime factorization Method:

☛ Break down all the numbers into their corresponding prime factors raised to their powers

☛ Select the distinct common prime factors with maximum power (among the factors of given numbers) and multiply all of them.

Eg. 2160, 3420

2160 = 8 x 27 x 10

= 23 x 33 x 21 x 51  

= 24 x 33 x 51

3420 = 2 x 171 x 10 = 2 x 9 x 19 x 10

=   21 x 32 x 191 x 21 x  51

=  22 x 32 x 51 x 191

 

Least Common Multiple = 24 x 33 x 51 x 191

= 41040

                                              

LCM x HCF (of two numbers) = Product of the two numbers

[This property can be applied only when we have 2 numbers]

Properties of HCF & LCM

☛ LCM of numbers is always divisible by HCF of numbers

☛ Difference between given numbers has to be divisible by HCF

If N is largest number which leaves the same remainder in dividing 3 different numbers – A, B and C,

then value of N = HCF [(middle number – smallest number), (largest number – middle number)]

Eg. Find the largest number which leaves the same remainder when dividing 49, 81 and 177

Soln:  Difference between 49 and 81 = 81-49 = 32

           Difference between 81 and 177 = 177 – 81 = 96

           HCF = 32

Therefore the highest number that will leave a common remainder is 32.

The same property can be extended to any number of given numbers

☛ If N is largest number which leaves the same remainder in dividing M different numbers,

value of N = HCF [(second smallest number – smallest number), (third smallest number- second smallest number), …(largest number – second largest number)]

☛ LCM finds applications in problems where different actions are repeated at some different time intervals and we are asked to find at what time, all of them will coincide.

PRACTICE PROBLEMS

1

HCF & LCM

1 / 35

The smallest fraction which each of  3/5 , 6/7, 5/16 will perfectly divide is:

2 / 35

What is the GCD (HCF) of 0.72, 0.054 and 0.0081?

3 / 35

The least multiple of 7, which leaves a remainder of 4, when divided by 5, 6, 9 and 12 is:

4 / 35

Let A be the smallest 5 digit number which leaves a remainder of 3 when divided by 6,7,8 or 9. Find the sum of digits of A

5 / 35

The greatest number of four digits which is divisible by 30, 50, 80 and 120 is:

6 / 35

Five bells keep ringing at intervals of 12 minutes,25 minutes, 60 minutes,30 minutes and 50 minutes respectively. If all the bells ring together at 3 PM, at which of the following time had all of them rung/ will ring together ?

7 / 35

Five bells commence tolling together and toll at intervals of 2, 4, 6, 8 and 12 seconds respectively. In 30 minutes, how many times do they toll together ?

8 / 35

Four clocks beep after 40 minutes, 1 hour 10 minutes, 1 hour 45 minutes and 2 hours respectively. If all the clocks had beeped together at 12 Noon, at what time will they beep again?

9 / 35

There are a certain number of soldiers in a field. If the soldiers are arranged in rows of 8 or 15 or 20, one soldier is left out. If the soldiers are arranged in rows of 11, there would be 10 soldiers left. What can be the number of soldiers in field?

10 / 35

A rectangular room is 2.99 m long and 4.83 m wide. Find the minimum number of square tiles with the same size that can be fitted so as to cover the entire flooring.

11 / 35

Prakash knows that the LCM and HCF of 2 numbers are 82944 and 2 respectively. If one of the numbers is 162, find the other.

12 / 35

Five bells toll at intervals of 3, 4, 12, 16 and 20 seconds respectively. If they toll together at the same time, after how many seconds will they toll together again, for the first time?

13 / 35

What would the value of the LCM of the three fractions - 1/14, 4/7 and 8/21 ?

14 / 35

If three numbers are in the ratio 3 : 4 : 5, and their LCM is 240, then find the sum of the three numbers.

15 / 35

Three different containers contain 425 litres, 475 litres and 525 litres. What is the biggest measure which can measure all the quantities accurately?

16 / 35

The sum of two numbers is 48 .Their HCF and LCM are 23 and 69 respectively. Find the square root of the sum of  reciprocals of two numbers

17 / 35

The HCF of two numbers is 64 and the numbers are in the ratio 13:17. What is the difference between those two numbers?

18 / 35

The H.C.F. of two numbers is 17 and the other two factors of their L.C.M. are 11 and 23. The smaller of the two numbers is:

19 / 35

The least number when divided by 36, 64 and 80 leave remainders of 26,54 and 70 respectively is:

20 / 35

Three number are in the ratio of 2 : 4 : 5 and their L.C.M. is 1200. Their H.C.F. is:

21 / 35

A, B and C start at the same time in the same direction to run around a circular stadium. A completes a round in 63 seconds, B in 45 seconds and c in 81 seconds, all starting at the same point. After what time (in seconds) will they again at the starting point ?

22 / 35

The least number which when divided by 7, 8, 9 and 10 leaves a remainder 3, but when divided by 100 leaves 43 as a remainder is:

23 / 35

The least number which should be added to 7287 so that the sum is exactly divisible by 5, 6, 8 and 10 is:

24 / 35

The greatest number which on dividing 1970 and 2870 leaves remainders 33 and 39 respectively, is:

25 / 35

The product of two co-prime numbers is 12319. Find the LCM of two numbers

26 / 35

The traffic lights at three different road crossings change after every 20 sec, 72 sec and 54 sec respectively. If they all change simultaneously at 4 : 10 : 00 pm, then find the time at which they will change simultaneously.

27 / 35

The HCF of two numbers is 37. Which of the following can never be the LCM of the two numbers?

28 / 35

LCM of two prime numbers x and y, where y>x is 899. Find the value of 4x -2y

29 / 35

Find the side of the largest square slab which can be paved on the floor of a room 1 m 53 cms long and 2 m 72 cm broad.

30 / 35

1001 pens and 770 rulers are to be distributed among students in such a way that, each student gets equal number of pens and rulers. The maximum number of students possible is:

31 / 35

Sachin, Virat and Dhoni start running at the same time in same direction around a circular lake. Sachin completes the whole round in 24 seconds, Virat in 20 seconds and Dhoni in 18 seconds, all starting at the same point. After what time will they meet again at starting point?

32 / 35

Find the number of common terms between the following two series 3,10, 17... 563 and 7,13,19,25...427

33 / 35

The HCF and LCM of two numbers are 31 and 2387 respectively. Find the difference between the two numbers.

34 / 35

What will be the least number which when tripled exactly divides 16,28,36 and 48?

35 / 35

The least square number which is divisible by each of 12,16, 48 and 60 is:

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