In this section, the following topics are discussed with examples

FACTORS AND MULTIPLES

If a number `m’ divides another number `n’ exactly, then we say that `m’ is a factor of `n’ and that `n’ is a multiple of `m’.

Eg:  3 is a factor of 12 and therefore 12 is a multiple of 3.

Interesting fact:

In math we have certain numbers whose powers form a triangle as the numbers increase,

i.e.,

    112= 121

    1112= 12321

    11112= 1234321

    111112= 123454321 and so on…

The concept is so simple as many number of 1’s that many number of consecutive natural numbers will come and there after it will decrease consecutively again till 1.

PRIME FACTORS

Numbers which has exactly two factors are called as prime numbers. Those numbers are namely 2,3,5,7,11,13,17,19,23,29,31,37,41,……So mostly when a number being factorized the numbers should be always written in terms of prime factors i.e., in a multiples of these above numbers. Because any other composite numbers can be derived using this prime factors. But in this also we must make sure about a context called as co-prime numbers.

CO - PRIME NUMBERS

A set of numbers which do not have any other common factor other than 1 are called co-prime or relatively prime numbers. This means those numbers whose HCF is 1.

For example, 8 and 9 have no other common factor other than 1 so they are co-prime numbers.

Properties Of Co-prime Numbers:

• All prime numbers are co-prime to each other.
• Any 2 consecutive integers are always co-prime.
• Sum of any two co-prime numbers is always co-prime with their product.
• 1 is co-prime with all numbers.
• a and b () are co-prime only if the numbers 2a-1 and 2b-1 are co-prime.

LEAST COMMON MULTIPLE (L.C.M.)

L.C.M. is the least non-zero number in common multiples of two or more numbers. 

Multiple of 6 = 6, 12, 18, 24, 30, …….. 

Multiple of 8 = 8, 16, 24, 32, 40, …….. 

Common Multiple of 6 and 8 = 24, 48 …………… 

Least Common Multiple = 24

Example 1:

Find the L.C.M. of 12, 27 and 40

Factors of 12 = 2x2x3 = 22×3

Factors of 27 =3x3x3=33

Factors of 40 = 2x2x2x5 = 23×5

HIGHEST COMMON FACTOR (H.C.F) :

The highest common factor of two or more numbers is the greatest number which divides each of them exactly.

Example 2

Find the H.C.F. of 24 and 56

Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24

Factors of 56 = 1, 2, 4, 7, 8, 14, 28, 56

Common factors of 24 and 56 are 1, 2, 4, 8 H.C.F. of 24 and 56 = 8 

Factorization Method:

H.C.F. can be found by resolving the given numbers into prime factors and then taking the product of least powers of all common factors, that occur in these numbers.

Properties of L.C.M & H.C.F.:

1. The product of two given numbers is equal to the product of their H.C.F. and L.C.M.

Product of numbers = H.C.F.of numbers / L.C.M. of two numbers

2. L.C.M. of given fractions = L.C.M.of numerators / H.C.F.of denominators

3. H.C.F of given fractions = H.C.F. of numerators / L.C.M.of denominators

4. The L.C.M of a given set of numbers would be either the highest or higher than the highest of the given numbers.

5.The H.C.F. of a given set of numbers would be either the lowest or lower than the lowest.

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.

ALBERT EINSTEIN

MORE EXAMPLES

Example 3

Find the L.C.M. of 125,64,8 and 3.
Sol:

          Given numbers are 125, 64, 8 and 3

          L.C.M. is 125x 64×3 = 24,000

Example 4

Find H.C.F. of 48, 108, 140 

Sol:

Factors of 48
                      = 2x2x2x2x3
                      = 24×3


Factors of 108

                      = 2x2x3x3x3

                      = 22×33

 

Factors of 140

                     =2x2x5x7

                     =22 x5x7

 

H.C.F.=22=4                  

Example 5

Find the L.C.M. of 1/3 , 5/6 , 5/9 , 10/27 ?  

Sol:

L.C.M. of fractions = L.C.M. of numerators / H.C.F.of denominators

 L.C.M. of 1,5 and 10 is 10

 H.C.F of 3,6,9 and 27 is 3

 L.C.M. of given fractions = 10 / 3

Example 6

Find the H.C.F. of 1/2, 3/4,5/6,7/8, 9/10 

Sol:

H.C.F. of fractions = H.C.F.of numerators / L.C.M. of denominators

H.C.F. of 1, 3, 5, 7 and 9 is 1

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

H.C.F. of given fractions = 1/120

Example 7

The L.C.M. of two number is 2310. Their H.C.F. is 30. If one number is 210, the other is: 

Sol:

The other number is 2310×30 = 210 330 

Example 8

Find the greatest number that will divide 197 and 269 and leaves 5 as reminder in each case.

Sol:

Required number  = H.C.F. of [(197-5) and (269-5)]

                                  = H.C.F. of (192 and 264) = 8

Example 9

Five bells begin to toll together and toll respectively at intervals of 6,7,8,9 and 12 seconds. How many times they will toll together in one hour, excluding the one at the start?

Sol:

                     L.C.M. of 6,7,8,9 and 12
                     = 2x2x3x7x2x3 = 504 

ie, The bells will toll together after each 504 seconds. In one hour, they will toll together 7 times.

Some important L.C.M and H.C.F tricks:

1) Product of two numbers = Their h.c.f. * Their l.c.m.

2) h.c.f. of given numbers always divides their l.c.m.

3) h.c.f. of given fractions =  h.c.f. of numerator

l.c.m. of denominator

4) l.c.m. of given fractions =   l.c.m. of numerator

h.c.f. of denominator

5) If d is the h.c.f. of two positive integer a and b, then there exist unique integer m and n, such that

d = am + bn

6) If p is prime and a,b are any integer then P ,This implies   P or P

ab a b

7) h.c.f. of a given number always divides its l.c.m.

Most important points about L.C.M and H.C.F problems :

 1) Largest number which divides x,y,z to leave same remainder = h.c.f. of y-x, z-y, z-x.

2) Largest number which divides x,y,z to leave remainder R (i.e. same) = h.c.f of x-R, y-R, z-R.

3) Largest number which divides x,y,z to leave same remainder a,b,c = h.c.f. of x-a, y-b, z-c.

4) Least number which when divided by x,y,z and leaves a remainder R in each case = ( l.c.m. of x,y,z) + R

Thus in this section we have learned about the concept of Factors, Multiples, Prime Factors, Co-Prime Numbers, L.C.M and H.C.F. In the next section, we will learn about Algebra.