HCF LCM FOR COMPETITIVE EXAMS

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HCF (Highest Common Factor):

The HCF of two or more than two numbers is the greatest number that divides each of them exactly.

Example: HCF of 6 & 16

6) 16 ( 2
12
——-
4) 6 ( 2
4
——-
2) 4 ( 2
4
——–
0

Therefore, HCF of 6 & 16 = 2
In brief,
6 = 1 × 6, 2 × 3
16 = 1× 16, 2 × 8
In both case, except 2, there is no other number divides exactly both numbers (6 & 16)

LCM (Least Common Multiple):

The least number is exactly divisible by each one of the given numbers. The least common multiple can be calculated manually as well as by using the LCM calculator. Below are a few examples of LCM.

Example: LCM of 12 & 16

2 LCM ( 12, 16 )
2 LCM ( 6, 4)
2 LCM ( 3, 2 )

LCM of 12 & 16 = 2 × 2 × 3 × 4
= 48
In brief
12 = 12 × 1, 12 × 2, 12 × 3, 12 × 4 (48) ….…….. 12 × 8 (96)

16 = 16 × 1, 16 × 2, 16 × 3 (48) …………………. 16 × 6 (96)

Here 48 is the least number which is exactly divisible by 12 & 16

**** Product of two numbers = product of their HCF & LCM (Formula)

HCF & LCM of fractions

HCF = HCF of Numerator/ LCM of Denominator
LCM = LCM of Numerator / HCF of Denominator

SHORT CUT TO FIND HCF ORALLY:

1. HCF of 8 & 12 = ?
Step 1: Difference of 8 & 12 is = 4
Step 2: 4 divides the 8 and 12 exactly

2. HCF of 21 & 35 = ?
Step 1: Difference of 21 & 35 = 14

= 2 × 7
Step 2: Here, 2 not divides the 21 & 35, therefore discards 2
Step 3: And 7 divides the 21 & 35 exactly

3. HCF of 27 & 32 = ?
Step 1: Difference of 27 & 32 = 5
Step 2: Here, 5 not divide both the given numbers exactly
So, HCF = 1

FIND OUT THE LCM JUST BY INSPECTION

Q. LCM of 5, 10, 25 & 50 = ?
Step 1: Here, 50 is the largest number and it is multiple of remaining numbers i. e 5, 10 & 25 divide the 50 exactly
So, LCM = Largest number = 50 (answer)

Q. LCM of 3, 9, 12 & 18 = ?
Step 1: Here, the highest number is 18, but 18 is not multiple of remaining numbers (i. e 3, 9 & 12)
Step 2: Now just double the 18, it becomes 18 × 2 = 36
Step 3: Now 36 is exactly divisible by 3, 9 & 12

Q. LCM of 2, 9, 13 & 18 = ?
Step 1: As we know that, LCM of 2, 9 & 18 = 18 (here, 18 is the largest number and it is exactly divisible by 2 & 9)
Step 2: Now 13 is a prime number when the prime number is given jut multiply the number i. e 18 × 13
So, LCM = 18 × 13 => 234 (answer)

Practice Problems

1 Find the HCF of 23 × 32 × 5 × 74, 22 × 35 × 52 ×73, 23 × 53 × 721

2 Find the LCM of 22 × 33 × 5 × 72, 23 × 32 × 52 × 74, 2 × 3 × 53 × 7 × 11

3 Find the HCF & LCM of 2/3 8/9 16/81 & 10/27

4 On dividing a number by 5, 6 & 7 we get 3, 4 & 5 as the remainder. Find the number

5 On dividing a number by 5, 6 & 7 we get 2 as remainder always, find that number

6 Find a number which after adding 7 is divisible by 10, 11 & 12

7 Two numbers are in the ratio 15: 11. If their HCF is 13. Find the numbers?

8 The ratio of two numbers is 3:4. If their LCM is 48. Find the numbers?

9 The HCF of the two numbers is 11 and their LCM is 693. If one of the numbers is 77, find the other?

10 The sum of the two numbers is 216 and their HCF is 27. The numbers are?

11 The product of the two numbers is 2028 and their HCF is 13. The number of such pairs is?

12 Find two 3- digit numbers whose LCM is 6188 and HCF is 68.

13 The HCF & LCM of the two numbers are 12 & 72 respectively. If the sum of the two numbers is 60. Find the two numbers.

14 The least number of soldiers that can be arranged in 12, 15 & 18 rows having an equal number of soldiers and can also be arranged in a solid square.

15 HCF & LCM of 77, 99 & x are 11 & 3465 respectively. The least value of x is ?

16 The maximum number of students among whom 1001 pens & 910 pencils can be distributed in such a way that each student gets the same number of pens & a same number of pencils is?

17 A room is 725 cm long and 575 cm wide. Its floor is paved with square tiles. What is the least number of tiles required?

18 Find the largest number which divides 62,132 & 237 to leave the same remainder in each case.

19 The sum of two numbers is 45 and their HCF & LCM are 3 & 168 respectively. The sum of the reciprocals of the numbers will be?

20 For what value of k, HCF of 2x2 + kx – 12 and x2 + x – 2k – 2 is x + 4 ?

21 If the LCM & HCF of two expressions are ( x2 + 6x + 8 )(x + 1) & (x + 1) respectively and one of the expressions is x2+3x+2, find the other.

22 In four consecutive prime numbers that are in ascending order, the product of the first three is 385 and that of the last three is 1001. The largest given prime number is?

23 There are five bells that start ringing together at intervals of 3, 6, 9, 12 & 15 seconds respectively. In 36 minutes, how many times will the bells ring simultaneously?

24 If A & B are the HCF & LCM respectively of two algebraic expressions x & y and A+B = x+ y, then the value of A3 + B3 is ?

25 The LCM of two positive integers is twice the larger number. The difference of the smaller number and the HCF of the two numbers is 4. The smaller number is?

26 A number lies between the cubes of 15 and 16. If the number is divisible by the square of 12 as well as by 7, what is the number?

27 The least number which is divisible by all the natural numbers up to and including 10 is….

28 Three numbers which are coprime to one another are such that the product of the first two is 551 and that of the last two is 1073. The sum of the three numbers is……

29 HCF and LCM of two numbers are 7 and 140 respectively. If the numbers are between 20 and 45, the sum of the numbers is ……..

30 The least five-digit perfect number which is divisible by 3, 4, 5 and 8 is

31 HCF of (4143 + 4343) and (4141 + 4341) is

32 When a heap of Pebbles is arranged into groups of 32 – each, 10 – Pebbles are leftover. When they are arranged in heaps of 40 – each, 18 – Pebbles are leftover and when in groups of 72 – each, 50 – are leftover. The least number of Pebbles in the heap is

Solutions for Practice Problems will be available on Fdaytalk Quantitative Aptitude Book, Download Here