\displaystyle \textbf{Factors and Multiples}
\displaystyle \text{If a number }a\text{ divides another number }b\text{ exactly, then }a\text{ is a factor of }b.
\displaystyle \text{Also, }b\text{ is a multiple of }a.
\displaystyle \textbf{Factor:}
\displaystyle \text{A factor of a number is an exact divisor of that number.}
\displaystyle \textbf{Multiple:}
\displaystyle \text{A number is said to be a multiple of each of its factors.}
\displaystyle \textbf{Example:}
\displaystyle \text{When }20\text{ is divided by }5,\text{ the remainder is }0.
\displaystyle \text{Thus, }5\text{ divides }20\text{ exactly.}
\displaystyle \therefore 5\text{ is a factor of }20\text{ and }20\text{ is a multiple of }5.
\displaystyle \\

\displaystyle \textbf{Even Numbers}
\displaystyle \text{All multiples of }2\text{ are called even numbers.}
\displaystyle \textbf{Example:}
\displaystyle 2,\ 4,\ 6,\ 8,\ 10,\ 12,\ldots\text{ are even numbers.}
\displaystyle \\

\displaystyle \textbf{Odd Numbers}
\displaystyle \text{Numbers which are not multiples of }2\text{ are called odd numbers.}
\displaystyle \textbf{Example:}
\displaystyle 1,\ 3,\ 5,\ 7,\ 9,\ 11,\ 13,\ 15,\ldots\text{ are odd numbers.}
\displaystyle \\

\displaystyle \textbf{Prime Numbers}
\displaystyle \text{A natural number greater than }1\text{ which has exactly two factors, }1\text{ and itself,}
\displaystyle \text{is called a prime number.}
\displaystyle \textbf{Example:}
\displaystyle 2,\ 3,\ 5,\ 7,\ 11,\ 13,\ 17,\ 19,\ldots\text{ are prime numbers.}
\displaystyle \\

\displaystyle \textbf{Composite Numbers}
\displaystyle \text{A natural number greater than }1\text{ which has more than two factors is called a composite number.}
\displaystyle \textbf{Example:}
\displaystyle 4,\ 6,\ 8,\ 9,\ 10,\ 12,\ 15,\ldots\text{ are composite numbers.}
\displaystyle \textbf{Note:}
\displaystyle \text{(a) }1\text{ is neither prime nor composite. It has only one factor, namely }1.
\displaystyle \text{(b) }2\text{ is the smallest prime number.}
\displaystyle \text{(c) }2\text{ is the only even prime number.}
\displaystyle \\

\displaystyle \textbf{Twin Primes}
\displaystyle \text{Two prime numbers whose difference is }2\text{ are called twin primes.}
\displaystyle \textbf{Examples:}
\displaystyle \text{(i) }3,\ 5\qquad\text{(ii) }5,\ 7\qquad\text{(iii) }11,\ 13
\displaystyle \\

\displaystyle \textbf{Prime Triplets}
\displaystyle \text{Three prime numbers of the form }p,\ p+2,\ p+4\text{ are called a prime triplet.}
\displaystyle \textbf{Example:}
\displaystyle \{3,\ 5,\ 7\}\text{ is the only prime triplet of this form.}
\displaystyle \\

\displaystyle \textbf{Perfect Numbers}
\displaystyle \text{A number is called a perfect number if the sum of all its positive factors is twice the number.}
\displaystyle \text{Equivalently, the sum of its proper positive factors is equal to the number itself.}
\displaystyle \textbf{Example 1:}
\displaystyle \text{The positive factors of }6\text{ are }1,\ 2,\ 3,\ 6.
\displaystyle 1+2+3+6=12=2\times6.
\displaystyle \therefore 6\text{ is a perfect number.}
\displaystyle \textbf{Example 2:}
\displaystyle \text{The positive factors of }28\text{ are }1,\ 2,\ 4,\ 7,\ 14,\ 28.
\displaystyle 1+2+4+7+14+28=56=2\times28.
\displaystyle \therefore 28\text{ is a perfect number.}
\displaystyle \\

\displaystyle \textbf{Co-Primes}
\displaystyle \text{Two positive integers are called co-primes if their only common factor is }1.
\displaystyle \text{Equivalently, two numbers are co-prime if their HCF is }1.
\displaystyle \textbf{Examples:}
\displaystyle \text{(i) }3,\ 4\qquad\text{(ii) }4,\ 9
\displaystyle \textbf{Note:}
\displaystyle \text{(a) Any two distinct prime numbers are always co-prime.}
\displaystyle \text{(b) Two co-prime numbers are not necessarily prime numbers.}
\displaystyle \\

\displaystyle \textbf{Test of Divisibility}
\displaystyle \textbf{Test of Divisibility by }3\textbf{:}
\displaystyle \text{A number is divisible by }3\text{ if the sum of its digits is divisible by }3.
\displaystyle \textbf{Examples:}
\displaystyle \text{Consider }814728.\text{ The sum of its digits is}
\displaystyle 8+1+4+7+2+8=30.
\displaystyle \text{Since }30\text{ is divisible by }3,\ 814728\text{ is divisible by }3.
\displaystyle \text{Consider }349709.\text{ The sum of its digits is}
\displaystyle 3+4+9+7+0+9=32.
\displaystyle \text{Since }32\text{ is not divisible by }3,\ 349709\text{ is not divisible by }3.
\displaystyle \\

\displaystyle \textbf{Test of Divisibility by }4\textbf{:}
\displaystyle \text{A number is divisible by }4\text{ if the number formed by its last two digits is divisible by }4.
\displaystyle \textbf{Examples:}
\displaystyle \text{Consider }23424224.\text{ The number formed by its last two digits is }24.
\displaystyle \text{Since }24\text{ is divisible by }4,\ 23424224\text{ is divisible by }4.
\displaystyle \text{Consider }234242241.\text{ The number formed by its last two digits is }41.
\displaystyle \text{Since }41\text{ is not divisible by }4,\ 234242241\text{ is not divisible by }4.
\displaystyle \\

\displaystyle \textbf{Test of Divisibility by }5\textbf{:}
\displaystyle \text{A number is divisible by }5\text{ if its units digit is either }0\text{ or }5.
\displaystyle \textbf{Examples:}
\displaystyle \text{Consider }234242245.\text{ Its units digit is }5.
\displaystyle \text{Hence, }234242245\text{ is divisible by }5.
\displaystyle \text{Consider }2342422451.\text{ Its units digit is }1.
\displaystyle \text{Hence, }2342422451\text{ is not divisible by }5.
\displaystyle \\

\displaystyle \textbf{Test of Divisibility by }6\textbf{:}
\displaystyle \text{A number is divisible by }6\text{ if it is divisible by both }2\text{ and }3.
\displaystyle \textbf{Example:}
\displaystyle \text{Consider }753222.\text{ Its units digit is }2,\text{ so it is divisible by }2.
\displaystyle \text{The sum of its digits is }7+5+3+2+2+2=21.
\displaystyle \text{Since }21\text{ is divisible by }3,\ 753222\text{ is also divisible by }3.
\displaystyle \therefore 753222\text{ is divisible by both }2\text{ and }3,\text{ and hence by }6.
\displaystyle \\

\displaystyle \textbf{Test of Divisibility by }8\textbf{:}
\displaystyle \text{A number is divisible by }8\text{ if the number formed by its last three digits is divisible by }8.
\displaystyle \textbf{Examples:}
\displaystyle \text{Consider }293512.\text{ The number formed by its last three digits is }512.
\displaystyle \text{Since }512\text{ is divisible by }8,\ 293512\text{ is divisible by }8.
\displaystyle \text{Consider }293513.\text{ The number formed by its last three digits is }513.
\displaystyle \text{Since }513\text{ is not divisible by }8,\ 293513\text{ is not divisible by }8.
\displaystyle \\

\displaystyle \textbf{Test of Divisibility by }9\textbf{:}
\displaystyle \text{A number is divisible by }9\text{ if the sum of its digits is divisible by }9.
\displaystyle \textbf{Examples:}
\displaystyle \text{Consider }874728.\text{ The sum of its digits is}
\displaystyle 8+7+4+7+2+8=36.
\displaystyle \text{Since }36\text{ is divisible by }9,\ 874728\text{ is divisible by }9.
\displaystyle \text{Consider }349709.\text{ The sum of its digits is}
\displaystyle 3+4+9+7+0+9=32.
\displaystyle \text{Since }32\text{ is not divisible by }9,\ 349709\text{ is not divisible by }9.
\displaystyle \\

\displaystyle \textbf{Test of Divisibility by }10\textbf{:}
\displaystyle \text{A number is divisible by }10\text{ if its units digit is }0.
\displaystyle \textbf{Examples:}
\displaystyle 10,\ 20,\ 3430,\ 23249860\text{ are all divisible by }10.
\displaystyle \\

\displaystyle \textbf{Test of Divisibility by }11\textbf{:}
\displaystyle \text{A number is divisible by }11\text{ if the difference between the sum of the digits at}
\displaystyle \text{alternate places is }0\text{ or a multiple of }11.
\displaystyle \textbf{Examples:}
\displaystyle \text{Consider }8192657.
\displaystyle \text{Sum of the digits at odd places from the right}=7+6+9+8=30.
\displaystyle \text{Sum of the digits at even places from the right}=5+2+1=8.
\displaystyle \text{The difference is }30-8=22.
\displaystyle \text{Since }22\text{ is divisible by }11,\ 8192657\text{ is divisible by }11.
\displaystyle \text{Consider }8192656.
\displaystyle \text{Sum of the digits at odd places from the right}=6+6+9+8=29.
\displaystyle \text{Sum of the digits at even places from the right}=5+2+1=8.
\displaystyle \text{The difference is }29-8=21.
\displaystyle \text{Since }21\text{ is not divisible by }11,\ 8192656\text{ is not divisible by }11.
\displaystyle \\

\displaystyle \textbf{Prime Factors}
\displaystyle \text{A factor of a given number is called a prime factor if it is a prime number.}
\displaystyle \textbf{Example:}
\displaystyle \text{The factors of }21\text{ are }1,\ 3,\ 7,\text{ and }21.
\displaystyle \text{Of these, }3\text{ and }7\text{ are prime numbers.}
\displaystyle \therefore 3\text{ and }7\text{ are the prime factors of }21.
\displaystyle \\

\displaystyle \textbf{Prime Factorization}
\displaystyle \text{The method of expressing a natural number greater than }1\text{ as a product of prime numbers}
\displaystyle \text{is called prime factorization or complete factorization of the given number.}
\displaystyle \textbf{Example:}
\displaystyle 60=2\times2\times3\times5=2^2\times3\times5.
\displaystyle \\

\displaystyle \textbf{Unique Factorization Property}
\displaystyle \text{Every natural number greater than }1\text{ can be expressed as a product of prime numbers}
\displaystyle \text{in a unique way, except for the order in which the prime factors are written.}
\displaystyle \textbf{Example:}
\displaystyle 30=2\times3\times5=3\times5\times2=5\times2\times3.
\displaystyle \text{The order may change, but the collection of prime factors remains the same.}
\displaystyle \\

2020-05-31_18-24-13

\displaystyle \therefore 36=2\times2\times3\times3=2^2\times3^2.
\displaystyle \\

\displaystyle \textbf{Common Factors}
\displaystyle \text{A number which divides each of the given numbers exactly is called a common factor}
\displaystyle \text{of the given numbers.}
\displaystyle \textbf{Example:}
\displaystyle \text{The factors of }36\text{ are }1,\ 2,\ 3,\ 4,\ 6,\ 9,\ 12,\ 18,\text{ and }36.
\displaystyle \text{The factors of }42\text{ are }1,\ 2,\ 3,\ 6,\ 7,\ 14,\ 21,\text{ and }42.
\displaystyle \therefore \text{The common factors of }36\text{ and }42\text{ are }1,\ 2,\ 3,\text{ and }6.
\displaystyle \\

\displaystyle \textbf{Highest Common Factor (H.C.F.) or Greatest Common Divisor (G.C.D.)}
\displaystyle \text{The H.C.F. or G.C.D. of two or more numbers is the greatest number that divides}
\displaystyle \text{each of the given numbers exactly.}
\displaystyle \textbf{Example:}
\displaystyle \text{Consider the numbers }36\text{ and }54.
\displaystyle \text{The factors of }36\text{ are }1,\ 2,\ 3,\ 4,\ 6,\ 9,\ 12,\ 18,\text{ and }36.
\displaystyle \text{The factors of }54\text{ are }1,\ 2,\ 3,\ 6,\ 9,\ 18,\ 27,\text{ and }54.
\displaystyle \text{The greatest common factor is }18.
\displaystyle \therefore \text{H.C.F.}(36,54)=18.
\displaystyle \\

\displaystyle \textbf{Methods of Finding the H.C.F. of Given Numbers}
\displaystyle \textbf{1. Prime Factorization Method}
\displaystyle \textbf{Step 1: }\text{Express each of the given numbers as a product of prime factors.}
\displaystyle \textbf{Step 2: }\text{Identify the prime factors common to all the given numbers.}
\displaystyle \textbf{Step 3: }\text{Take the least power of each common prime factor and multiply them.}
\displaystyle \text{The resulting product is the H.C.F. of the given numbers.}
\displaystyle \textbf{Example:}
\displaystyle \text{Find the H.C.F. of }324,\ 288\text{ and }360.
\displaystyle \text{First, express each number as a product of prime factors.}
\displaystyle 324=2^2\times3^4
\displaystyle 288=2^5\times3^2
\displaystyle 360=2^3\times3^2\times5
\displaystyle \text{The common prime factors are }2\text{ and }3.
\displaystyle \text{The least power of }2\text{ is }2^2\text{ and the least power of }3\text{ is }3^2.
\displaystyle \therefore \text{H.C.F.}=2^2\times3^2=4\times9=36.
\displaystyle \\

\displaystyle \textbf{2. Long Division Method}
\displaystyle \textbf{Step 1: }\text{Divide the larger number by the smaller number.}
\displaystyle \textbf{Step 2: }\text{If the remainder is not }0,\text{ divide the previous divisor by the remainder.}
\displaystyle \textbf{Step 3: }\text{Continue dividing the preceding divisor by the remainder obtained}
\displaystyle \text{until the remainder becomes }0.
\displaystyle \textbf{Step 4: }\text{The last non-zero divisor is the H.C.F. of the given numbers.}
\displaystyle \\

2020-05-31_18-25-572020-05-31_18-26-34

\displaystyle \textbf{Applications of H.C.F.}
\displaystyle \textbf{1. Co-Prime Numbers:}
\displaystyle \text{Two natural numbers are said to be co-prime if their H.C.F. is }1.
\displaystyle \textbf{2. Reducing a Fraction to its Simplest Form:}
\displaystyle \text{To reduce a fraction }\frac{p}{q}\text{ to its simplest form, divide its numerator and denominator}
\displaystyle \text{by the H.C.F. of }p\text{ and }q,\text{ where }q\ne0.
\displaystyle \\

\displaystyle \textbf{Least Common Multiple (L.C.M.)}
\displaystyle \text{The L.C.M. of two or more natural numbers is the least natural number which is a multiple}
\displaystyle \text{of each of the given numbers.}
\displaystyle \textbf{Example:}
\displaystyle \text{Consider the numbers }12\text{ and }18.
\displaystyle \text{Multiples of }12=12,\ 24,\ 36,\ 48,\ldots
\displaystyle \text{Multiples of }18=18,\ 36,\ 54,\ 72,\ldots
\displaystyle \text{The least common multiple is }36.
\displaystyle \therefore \text{L.C.M.}(12,18)=36.
\displaystyle \\

\displaystyle \textbf{Methods of Finding the L.C.M. of Given Numbers}
\displaystyle \textbf{1. Prime Factorization Method}
\displaystyle \textbf{Step 1: }\text{Express each of the given numbers as a product of prime factors.}
\displaystyle \textbf{Step 2: }\text{Identify all the prime factors occurring in the prime factorizations.}
\displaystyle \textbf{Step 3: }\text{Take the highest power of each prime factor and multiply them.}
\displaystyle \text{The resulting product is the L.C.M. of the given numbers.}
\displaystyle \textbf{Example:}
\displaystyle \text{Find the L.C.M. of }180,\ 216\text{ and }324.
\displaystyle \text{First, express each number as a product of prime factors.}
\displaystyle 180=2^2\times3^2\times5
\displaystyle 216=2^3\times3^3
\displaystyle 324=2^2\times3^4
\displaystyle \text{The highest power of }2\text{ is }2^3,\text{ of }3\text{ is }3^4,\text{ and of }5\text{ is }5.
\displaystyle \therefore \text{L.C.M.}=2^3\times3^4\times5
\displaystyle =8\times81\times5=3240.
\displaystyle \\

\displaystyle \textbf{2. Common Division Method}
\displaystyle \textbf{Step 1: }\text{Write the given numbers in a row.}
\displaystyle \textbf{Step 2: }\text{Divide by a prime number that divides at least one of the given numbers exactly.}
\displaystyle \text{Carry forward unchanged any number that is not divisible by that prime number.}
\displaystyle \textbf{Step 3: }\text{Repeat the process until all the quotients become }1.
\displaystyle \textbf{Step 4: }\text{The product of all the prime divisors used is the required L.C.M.}
\displaystyle \\

2020-05-31_18-26-59

\displaystyle \therefore \text{L.C.M.}=2\times2\times2\times3\times2\times3=144.
\displaystyle \\

\displaystyle \textbf{Relation between H.C.F. and L.C.M. of Two Numbers}
\displaystyle \textbf{1. Product of Two Numbers:}
\displaystyle \text{For any two positive integers }a\text{ and }b,
\displaystyle a\times b=\text{H.C.F.}(a,b)\times\text{L.C.M.}(a,b).
\displaystyle \text{Thus, the product of two numbers is equal to the product of their H.C.F. and L.C.M.}
\displaystyle \\

\displaystyle \textbf{2. H.C.F. and L.C.M. of Fractions}
\displaystyle \textbf{(a) H.C.F. of Fractions:}
\displaystyle \text{H.C.F. of the given fractions}
\displaystyle =\frac{\text{H.C.F. of the numerators}}{\text{L.C.M. of the denominators}}.
\displaystyle \textbf{(b) L.C.M. of Fractions:}
\displaystyle \text{L.C.M. of the given fractions}
\displaystyle =\frac{\text{L.C.M. of the numerators}}{\text{H.C.F. of the denominators}}.
\displaystyle \\

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