DIVISIORS OF NATURAL NUMBER

STANDARD FORM OF NUMBER:

A number N >1 is said to be expressed in the standard form N = p^a. q^b.r^c………

Where p, q, r… are distinct primes and a, b, c… are positive integers.

EXAMPLE:

Standard form of 36 = 2² x 3²

Standard form of 120 = 2³ x 3¹ x 5¹

The number of divisors of a natural number

The number of divisors of a natural number N is denoted by d (N).

Steps for finding the number of divisors of a natural number.

Step 1: Write the given natural number in standard form i.e. N = p^a. q^b.r^c………

Step 2: d (N) = (a+1) (b+1)(c+1)…

EXAMPLE:

1. Find the number of divisors of 36.

36 = 2² x 3²

d (36) = (2+1)(2+1)

= 3 x 3

= 9.

2. Find the number of divisors of 120.

120 = 2³ x 3¹ x 5¹

d (120) = (3+1)(1+1)(1+1)

= 4 x 2 x 2

= 16.

The number of odd divisors of a natural number

Steps for finding the number odd divisors of a natural number.

Step 1: write the number in standard form.

Step 2: Then, consider only the exponents of odd prime factors.

Step 3: Add 1 to each of these exponents and multiply the results together.

Step 4: The final product is the number of odd divisors.

EXAMPLE:

1. Find the number of odd divisors of 36:

Standard form: 36 = 2² x 3²

Consider odd prime factors: 3²

Add 1 to each odd exponent: (2 + 1) = 3

No. of odd divisors of 36 = 3

2. Find the number of odd divisors of 120:

Prime Factorization: 120 = 2³ x 3¹ x 5¹

Consider odd prime factors: 3¹ and 5¹

Add 1 to each odd exponent and multiply them: (1 +1) (1+1)

No. of odd divisors of 120 = 2 x 2

= 4.

The number of even divisors of a natural number.

Number of Even Divisors = Number of Total Divisors – Number of Odd Divisors

EXAMPLE:

1. Find the number of even divisors of 36:

Number of total divisors = 9

Number of odd divisors = 3

Number of even divisors = 9-3 =6

FOR MORE UNDERSTANDING

NUBER	DIVISORS	ODD DIVISORS	EVEN DIVISORS
36	1, 2, 3, 4, 6, 9, 12,18, 36	1, 3, 9	2, 4, 6, 12, 18, 36
TOTAL	9	3	6

worksheet pdf given below

DIVISIORS OF NATURAL NUMBER