I.Factors and Multiples : If a number 'a' divides another number 'b' exactly, we say that 'a' is a factor of 'b'. In this case, b is called a multiple of a.


II.Highest Common Factor (H.C.F) or Greatest Common Measure (G.C.M) or Greatest Common Divisor (G.C.D) : The H.C.F. of two or more than two numbers is the greatest number that divides each of them exactly. There are two methods of finding the H.C.F. of a given set of numbers :

1. Factorization Method : Express each one of the given numbers as the product of prime factors.The product of least powers of common prime factors gives H.C.F.

2. Division Method : Suppose we have to find the H.C.F. of two given numbers. Divide the larger number by the smaller one. Now, divide the divisor by the remainder. Repeat the process of dividing the preceding number by the remainder last obtained till zero is obtained as remainder. The last divisor is the required H.C.F.

Finding the H.C.F. of more than two numbers : Suppose we have to find the H.C.F. of three numbers. Then, H.C.F. of [(H.C.F. of any two) and (the third number)] gives the H.C.F. of three given numbers. Similarly, the H.C.F. of more than three numbers may be obtained. 


III.Least Common Multiple (L.C.M) : The least number which is exactly divisible by each one of the given numbers is called their L.C.M.

1. Factorization Method of Finding L.C.M: Resolve each one of the given numbers into a product of prime factors. Then, L.C.M. is the product of highest powers of all the factors.

2. Common Division Method (Short-cut Method) of Finding L.C.M : Arrange the given numbers in a row in any order. Divide by a number which divides exactly at least two of the given numbers and carry forward the numbers which are not divisible. Repeat the above process till no two of the numbers are divisible by the same number except 1. The product of the divisors and the undivided numbers is the required L.C.M. of the given numbers.


IV. Product of two numbers = Product of their H.C.F and L.C.M


V. Co-primes : Two numbers are said to be co-primes if their H.C.F. is 1.


VI. H.C.F and L.C.M of Fractions :

1. H.C.F = H.C.F. of Numerators / L.C.M of Numerators

2. L.C.M = L.C.M of Numerators / H.C.F of Denominators


VII. H.C.F and L.C.M of Decimal Fractions : In given numbers, make the same number of decimal places by annexing zeros in some numbers, if necessary. Considering these numbers without decimal point, find H.C.F. or L.C.M. as the case may be. Now, in the result, mark off as many decimal places as are there in each of the given numbers.


VIII. Comparison of Fractions : Find the L.C.M. of the denominators of the given fractions. Convert each of the fractions into an equivalent fraction with L.C.M. as the denominator, by multiplying both the numerator and denominator by the same number. The resultant fraction with the greatest numerator is the greatest.


Three number are in the ratio of 3 : 4 : 5 and their L.C.M. is 2400. Their H.C.F. is:

A) 40 B) 80
C) 120 D) 200
Answer & Explanation Answer: A) 40


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.

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520 151036

If the sum of two numbers is 55 and the H.C.F. and L.C.M. of these numbers are 5 and 120 respectively, then the sum of the reciprocals of the numbers is equal to:

A) 55/601 B) 601/55
C) 11/120 D) 120/11
Answer & Explanation Answer: C) 11/120


Let the numbers be a and b.


Then, a + b = 55 and ab = 5 x 120 = 600.


The required sum =1a+1b = a+bab55600=11120

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183 111593

The L.C.M of  22, 54, 108, 135 and 198 is


A) 330 B) 1980
C) 5940 D) 11880
Answer & Explanation Answer: C) 5940


 22 = 2 x 11

 54 = 2×33

108 = 22×33

135 = 33×5 

198 = 2×32×11

  L.C.M = 22×33×5×11=5940

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286 92343

A rectangular courtyard 3.78 meters long 5.25 meters wide is to be paved exactly with square  tiles, all of the same size. what is the largest size of the tile which could be used for the purpose?

A) 14 cms B) 21 cms
C) 42 cms D) None of these
Answer & Explanation Answer: B) 21 cms


3.78 meters =378 cm = 2 × 3 × 3 × 3 × 7

5.25 meters=525 cm = 5 × 5 × 3 × 7

Hence common factors are 3 and 7

Hence LCM = 3 × 7 = 21

Hence largest size of square tiles that can be paved exactly with square tiles is 21 cm.

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198 86688

The product of two numbers is 2028 and their H.C.F. is 13. The number of such pairs is:

A) 1 B) 2
C) 3 D) 4
Answer & Explanation Answer: B) 2


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.

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249 70321

The L.C.M of two numbers is 495 and their H.C.F is 5. If the sum of the numbers is 100, then their difference is 

A) 10 B) 46
C) 70 D) 90
Answer & Explanation Answer: A) 10


Let the numbers be x and (100-x).




 =>  x2-100x+2475=0


 =>  (x-55) (x-45) = 0


 =>  x = 55 or x = 45


  The numbers are 45 and 55


Required difference = (55-45) = 10

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157 64513

L.C.M of two prime numbers x and y (x>y) is 161. The value of 3y-x is :

A) -2 B) -1
C) 1 D) 2
Answer & Explanation Answer: A) -2


 H. C. F of two prime numbers is 1.  Product of numbers = 1 x 161 = 161.

 Let the numbers be a and b . Then , ab= 161.

 Now, co-primes with  product 161 are (1, 161) and (7, 23).

 Since x and y are prime numbers and x >y , we have x=23 and y=7.

 Therefore,  3y-x = (3 x 7)-23 = -2

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144 64281

The product of two numbers is 4107. If the H.C.F of these numbers is 37, then the greater number is 

A) 101 B) 107
C) 111 D) 185
Answer & Explanation Answer: C) 111


 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).

 Therefore, Greater number = 111.

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85 61702