Hcf And Lcm

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 HCF & LCM

HCF & LCM are acronym for words, Highest common factor and Lowest common multiple respectively.

1. H. C. F

While we all know what a multiplication is like 2 * 3 = 6. HCF is just the reverse of multiplication which is known as Factorization. Now factorization is breaking a composite number into its prime factors. Like 6 = 2 * 3, where 6 is a composite number and 2 & 3 are prime number.

“In mathematics, the Highest Common Factor (HCF) of two or more integers is the largest positive integer that divides the numbers without a remainder. For example, the HCF of 8 and 12 is 4.”

Calculation

– By Prime Factorizations

Highest Common Factor can be calculated by first determining the prime factors of the two numbers and then comparing those factors, to take out the common factors.

As in the following example: HCF (18, 42), we find the prime factors of 18 = 2 * 3 * 3 and 42 = 7 * 2 * 3 and notice the “common” of the two expressions is 2 * 3; So HCF (18, 42) = 6.

– By Division Method

In this method first divide a higher number by smaller number.

• Put the higher number in place of dividend and smaller number in place of divisor.
• Divide and get the remainder then use this remainder as divisor and earlier divisor as dividend.
• Do this until you get a zero remainder. The last divisor is the HCF.
• If there are more than two numbers then we continue this process as we divide the third lowest number by the last divisor obtained in the above steps.

First find H.C.F. of 72 and 126

72|126|1
72       
54| 72|1
              54
              18| 54| 3
                    54
                      0 

H.C.F. of 72 and 126 = 18

2. L.C.M

The Least Common Multiple of two or more integers is always divisible by all the integers it is derived from.  For example, 20 is a multiple of 5 because 5 × 4 = 20, so 20 is divisible by 5 and 2. Because 10 is the smallest positive integer that is divisible by both 5 and 2, it is the least common multiple of 5 and 4.

LCM cam also be understand by this example:

Multiples of 5 are:

5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70 …

And the multiples of 6 are:

6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, …

Common multiples of 5 and 6 are:

30, 60, 90, 120, ….

Hence, the lowest common multiple is simply the first number in the common multiple list i.e 30.

Calculation

– By Prime Factorizations

The prime factorization theorem says that every positive integer greater than 1 can be written in only one way as a product of prime numbers.

Example: To find the value of LCM (9, 48, and 21).

First, find the factor of each number and express it as a product of prime number powers.

Like 9 = 3²,

48 = 2⁴ * 3

21 = 3 * 7

Then, write all the factors with their highest power like 3², 2⁴, and 7. And multiply them to get their LCM.

Hence, LCM (9, 21, and 48) is 3² * 2⁴ * 7 = 1008.

– By Division Method

Here, divide all the integers by a common number until no two numbers are further divisible. Then multiply the common divisor and the remaining number to get the LCM.

2 | 72, 240, 196
  2 | 36, 120, 98
   2 | 18, 60, 49
  3 | 9, 30, 49
      | 3, 10, 49

L.C.M. of the given numbers
= product of divisors and the remaining numbers
= 2×2×2×3×3×10×49
= 72×10×49 = 35280.

Relation between L.C.M. and H.C.F. of two natural numbers

The product of L.C.M. and H.C.F. of two natural numbers = the product of the numbers.

For Example:

LCM (8, 28) = 56 & HCF (8, 28) = 4

Now, 8 * 28 = 224 and also, 56 * 4 = 224

HCF & LCM of FRACTIONS:

Formulae for finding the HCF & LCM of a fractional number.

HCF of fraction = HCF of numerator / LCM of denominator

LCM of Fraction = LCM of Numerator / HCF of Denominator

Questions:

Level-I:

Level-II:

Answers:

Level-I:

Answer:1 Option A

Explanation:

Required number = H.C.F. of (91 – 43), (183 – 91) and (183 – 43)

     = H.C.F. of 48, 92 and 140 = 4.

Answer:2 Option C

Explanation:

Clearly, the numbers are (23 x 13) and (23 x 14).

• Larger number = (23 x 14) = 322.

Answer:3 Option D

Explanation:

L.C.M. of 2, 4, 6, 8, 10, 12 is 120.

So, the bells will toll together after every 120 seconds(2 minutes).

Answer:4 Option A

Explanation:

N = H.C.F. of (4665 – 1305), (6905 – 4665) and (6905 – 1305)

  = H.C.F. of 3360, 2240 and 5600 = 1120.

Sum of digits in N = ( 1 + 1 + 2 + 0 ) = 4

Answer:5 Option C

Explanation:

Greatest number of 4-digits is 9999.

L.C.M. of 15, 25, 40 and 75 is 600.

On dividing 9999 by 600, the remainder is 399.

 Required number (9999 – 399) = 9600.

Answer:6 Option C

Explanation:

Let the numbers be 37a and 37b.

Then, 37a x 37b = 4107

 ab = 3.

Now, co-primes with product 3 are (1, 3).

So, the required numbers are (37 x 1, 37 x 3) i.e., (37, 111).

 Greater number = 111.

Answer:7 Option A

Explanation:

Let the numbers be 3x, 4x and 5x.

Then, their L.C.M. = 60x.

So, 60x = 2400 or x = 40.

 The numbers are (3 x 40), (4 x 40) and (5 x 40).

Hence, required H.C.F. = 40.

Answer:8 Option C

Explanation:

Given numbers are 1.08, 0.36 and 0.90.   H.C.F. of 108, 36 and 90 is 18,

 H.C.F. of given numbers = 0.18.

Answer:9 Option B

Explanation:

Let the numbers 13a and 13b.

Then, 13a x 13b = 2028

 ab = 12.

Now, the co-primes with product 12 are (1, 12) and (3, 4).

[Note: Two integers a and b are said to be coprime or relatively prime if they have no common positive factor other than 1 or, equivalently, if their greatest common divisor is 1 ]

So, the required numbers are (13 x 1, 13 x 12) and (13 x 3, 13 x 4).

Clearly, there are 2 such pairs.

Answer:10 Option D

Explanation:

L.C.M. of 6, 9, 15 and 18 is 90.

Let required number be 90k + 4, which is multiple of 7.

Least value of k for which (90k + 4) is divisible by 7 is k = 4.

 Required number = (90 x 4) + 4   = 364.

Level-II:

Answer:11 Option C

Explanation:

 2 | 24  –  36  – 40

 ——————–

 2 | 12  –  18  – 20

 ——————–

 2 |  6  –   9  – 10

 ——————-

 3 |  3  –   9  –  5

 ——————-

   |  1  –   3  –  5

L.C.M.  = 2 x 2 x 2 x 3 x 3 x 5 = 360.

Answer:12 Option C

Explanation:

L.C.M. of 5, 6, 4 and 3 = 60.

On dividing 2497 by 60, the remainder is 37.

 Number to be added = (60 – 37) = 23.

Answer:13 Option C

Explanation:

 128352) 238368 ( 1

         128352

         —————

         110016 ) 128352 ( 1

                  110016

                 ——————  

                   18336 ) 110016 ( 6       

                           110016

                           ——-

                                x

                           ——-

 So, H.C.F. of 128352 and 238368 = 18336.

             128352     128352 ÷ 18336    7

 Therefore,  ——  =  ————– =  —

             238368     238368 ÷ 18336    13  

Answer:14 Option B

Explanation:

L.C.M. of 5, 6, 7, 8 = 840.

 Required number is of the form 840k + 3

Least value of k for which (840k + 3) is divisible by 9 is k = 2.

 Required number = (840 x 2 + 3) = 1683.

Answer:15 Option D

Explanation:

L.C.M. of 252, 308 and 198 = 2772.

So, A, B and C will again meet at the starting point in 2772 sec. i.e., 46 min. 12 sec.

Answer:16 Option C

Explanation:

Answer:17 Option B

Explanation:

 L.C.M. of 12, 18, 21 30                 2 | 12  –  18  –  21  –  30

                                         —————————-

   = 2 x 3 x 2 x 3 x 7 x 5 = 1260.       3 |  6  –   9  –  21  –  15

                                         —————————-

   Required number = (1260 ÷ 2)            |  2  –   3  –   7  –   5

                   = 630.

Answer:18 Option D

Explanation:

Let the numbers be 3x and 4x. Then, their H.C.F. = x. So, x = 4.

So, the numbers 12 and 16.

L.C.M. of 12 and 16 = 48.

Answer19: Option B

Explanation:

Required number = (L.C.M. of 12,16, 18, 21, 28) + 7

   = 1008 + 7

   = 1015

Answer:20 Option A

Explanation:

Clearly, 252 = 2 x 2 x 3 x 3 x 7.

,

Introduction to HCF and LCM

The greatest common factor (GCF) or highest common factor (HCF) of two or more integers is the largest number that is a factor of both integers. For example, the GCF of 12 and 18 is 6, because 6 is the largest number that is a factor of both 12 and 18.

The least common multiple (LCM) of two or more integers is the smallest number that is a multiple of both integers. For example, the LCM of 12 and 18 is 36, because 36 is the smallest number that is a multiple of both 12 and 18.

Finding HCF and LCM by prime factorization

The prime factorization of a number is the expression of that number as a product of prime numbers. For example, the prime factorization of 12 is $2\times2\times3$, and the prime factorization of 18 is $2\times3\times3$.

To find the GCF of two numbers, we can find the prime factorization of each number and then find the factors that are common to both prime factorizations. For example, the prime factorization of 12 is $2\times2\times3$, and the prime factorization of 18 is $2\times3\times3$. The factors that are common to both prime factorizations are $2$ and $3$, so the GCF of 12 and 18 is $6$.

To find the LCM of two numbers, we can find the prime factorization of each number and then multiply together all the prime factors, including each prime factor as many times as it appears in the prime factorizations. For example, the prime factorization of 12 is $2\times2\times3$, and the prime factorization of 18 is $2\times3\times3$. The LCM of 12 and 18 is therefore $2\times2\times3\times3=36$.

Finding HCF and LCM by division method

The division method is a way to find the GCF and LCM of two numbers by repeatedly dividing one number by the other. To find the GCF, we start by dividing the larger number by the smaller number. If the remainder is 0, then the smaller number is the GCF. If the remainder is not 0, we then divide the smaller number by the remainder. We continue this process until the remainder is 0. The last number that we divide by is the GCF.

To find the LCM, we start by dividing the larger number by the smaller number. We then multiply the quotient by the smaller number. We continue this process until the product is a multiple of both numbers. The last product that we get is the LCM.

Finding HCF and LCM by using tables

We can also find the GCF and LCM of two numbers by using tables. To find the GCF, we first make a table of the prime factors of each number. We then find the factors that are common to both numbers. The product of the common factors is the GCF.

To find the LCM, we first make a table of the prime factors of each number. We then find the factors that are in each number. We multiply together all the factors, including each factor as many times as it appears in the table. The product of the factors is the LCM.

Applications of HCF and LCM

The GCF and LCM of two numbers have many applications in mathematics and everyday life. For example, the GCF can be used to find the greatest common divisor of two polynomials. The LCM can be used to find the least common multiple of two fractions. The GCF and LCM can also be used to solve problems involving fractions, decimals, and percentages.

Problems on HCF and LCM

Here are some problems on HCF and LCM:

1. Find the GCF of 12 and 18.
2. Find the LCM of 12 and 18.
3. Find the GCF of 12, 18, and 24.
4. Find the LCM of 12, 18, and 24.
5. Find the GCF of 12, 18, 24, and 36.
6. Find the LCM of 12, 18, 24, and 36.
7. Find the GCF of 12, 18, 24, 36, and 48.

Factors

A factor of a number is a whole number that can be divided evenly into that number. For example, 1, 2, 3, 4, 6, and 12 are all factors of 12.

The factors of a number can be found by listing all the whole numbers that divide evenly into that number.

Prime numbers

A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number.

The first 25 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, and 83.

Greatest common factor (GCF)

The greatest common factor (GCF) of two or more numbers is the largest number that is a factor of all of the numbers.

For example, the GCF of 12 and 18 is 6.

The GCF of two or more numbers can be found by listing all the factors of each number and then finding the number that is common to all of the lists.

Least common multiple (LCM)

The least common multiple (LCM) of two or more numbers is the smallest number that is a multiple of all of the numbers.

For example, the LCM of 12 and 18 is 36.

The LCM of two or more numbers can be found by multiplying the numbers together and then dividing the product by their GCF.

Divisibility rules

Divisibility rules are a set of rules that can be used to determine whether a number is divisible by another number.

For example, the divisibility rule for 3 is that a number is divisible by 3 if the sum of its digits is divisible by 3.

The divisibility rules for the other common numbers are as follows:

• Divisibility rule for 2: A number is divisible by 2 if the last digit of the number is even.
• Divisibility rule for 5: A number is divisible by 5 if the last digit of the number is 0 or 5.
• Divisibility rule for 7: A number is divisible by 7 if the difference of the sum of the digits at odd places and the sum of the digits at even places is divisible by 7.
• Divisibility rule for 11: A number is divisible by 11 if the difference of the sum of the digits at odd places and the sum of the digits at even places, starting from the right, is divisible by 11.
• Divisibility rule for 13: A number is divisible by 13 if the sum of the digits of the number, when divided by 13, leaves a remainder of 0.
• Divisibility rule for 17: A number is divisible by 17 if the difference of the sum of the digits at odd places and the sum of the digits at even places, starting from the right, is divisible by 17.

Prime factorization

The prime factorization of a number is the expression of that number as a product of prime numbers.

For example, the prime factorization of 12 is 2 x 2 x 3.

The prime factorization of a number can be found by listing all the prime factors of the number and then multiplying them together.

Exponents

An exponent is a number that indicates how many times a base number is multiplied by itself.

For example, in the expression 23, the base number is 2 and the exponent is 3. This means that 2 is multiplied by itself 3 times.

The value of an expression with an exponent can be found by multiplying the base number by itself the number of times indicated by the exponent.

For example, the value of 23 is 2 x 2 x 2 = 8.

Base-ten system

The base-ten system is a Number System that uses the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 to represent numbers.

In the base-ten system, each digit has a place value that is determined by its position in the number.

The place value of a digit is the product of the digit and the value of

Sure, here are some MCQs without mentioning the topic HCF and LCM:

1. What is the greatest common factor of 12 and 18?
   (A) 6
   (B) 9
   (C) 12
   (D) 18

2. What is the least common multiple of 12 and 18?
   (A) 6
   (B) 9
   (C) 12
   (D) 18

3. What is the greatest common factor of 12, 18, and 24?
   (A) 6
   (B) 9
   (C) 12
   (D) 18

4. What is the least common multiple of 12, 18, and 24?
   (A) 6
   (B) 9
   (C) 12
   (D) 18

5. What is the greatest common factor of 24, 36, and 48?
   (A) 12
   (B) 18
   (C) 24
   (D) 36

6. What is the least common multiple of 24, 36, and 48?
   (A) 12
   (B) 18
   (C) 24
   (D) 36

7. What is the greatest common factor of 36, 48, and 60?
   (A) 12
   (B) 18
   (C) 24
   (D) 36

8. What is the least common multiple of 36, 48, and 60?
   (A) 12
   (B) 18
   (C) 24
   (D) 36

9. What is the greatest common factor of 48, 60, and 72?
   (A) 12
   (B) 18
   (C) 24
   (D) 36

10. What is the least common multiple of 48, 60, and 72?
    (A) 12
    (B) 18
    (C) 24
    (D) 36

11. What is the greatest common factor of 60, 72, and 84?
    (A) 12
    (B) 18
    (C) 24
    (D) 36

12. What is the least common multiple of 60, 72, and 84?
    (A) 12
    (B) 18
    (C) 24
    (D) 36

13. What is the greatest common factor of 72, 84, and 96?
    (A) 12
    (B) 18
    (C) 24
    (D) 36

14. What is the least common multiple of 72, 84, and 96?
    (A) 12
    (B) 18
    (C) 24
    (D) 36

15. What is the greatest common factor of 84, 96, and 108?
    (A) 12
    (B) 18
    (C) 24
    (D) 36

16. What is the least common multiple of 84, 96, and 108?
    (A) 12
    (B) 18
    (C) 24
    (D) 36

17. What is the greatest common factor of 96, 108, and 120?
    (A) 12
    (B) 18
    (C) 24
    (D) 36

18. What is the least common multiple of 96, 108, and 120?
    (A) 12
    (B) 18
    (C) 24
    (D) 36

19. What is the greatest common factor of 108, 120, and 132?
    (A) 12
    (B) 18
    (C) 24
    (D) 36

20. What is the least common multiple of 108,