Sample Problems on Divisors

Problem 1: Find all the Divisors of 14.

Solution:

• 1 x 14 = 14
• 2 x 7 = 14
• 7 x 2 = 14

The divisors of the number 14 are 1, 2, 7 and 14

Therefore, the number of divisors of 14 is 4.

Problem 2: Find all the Divisors of 24

Solution:

• 1 x 24 = 24
• 2 x 12 = 24
• 3 x 8 = 24
• 4 x 6 = 24
• 6 x 4 = 24

The factors of the number 24 are 1, 2, 3, 4, 6 and 24

Therefore, the number of divisors of 24 is 6.

Problem 3: Find prime divisors of 460

Solution:

Step 1: Take the number 460 and and divide it with the smallest prime factor 2

• 460 ÷ 2 = 230

Step 2: Again divide 230 by the smallest prime factor 2 as 230 is divisible by 2

• 230 ÷ 2 = 115

Step 3: Now, 115 is not completely divisible by 2 as it will give a fractional number. Let’s divide it with the next smallest prime factor 5

• 115 ÷ 5 = 23

Step 4: As 23 is itself a prime number as it can only be divided by itself and 1. We cannot proceed further.

Therefore, the prime divisors of number 460 are 2² x 5 x 23

Problem 4: Find the number of divisors of 48.

Solution:

It is calculated using the following formula:

d(n) = (x₁ + 1) (x₂ + 1) . . . (x_k + 1)

The prime factorization of 48 is 2⁴ x 3. Therefore, the number of divisors of 12 are:

⇒ d(48) = (4 +1) (1+1)

⇒ d(48) = 5 x 2

⇒ d(48) = 10

Therefore, there are 10 divisors of 48

Problem 5: Find the number of divisors of 1080

Solution:

It is calculated using the following formula:

d(n) = (x₁ + 1) (x₂ + 1) . . . (x_k + 1)

The prime factorization of 1080 is 2³ x 3³ x 5. Therefore, the number of divisors of 1080 are:

⇒ d(1080) = (3+1) (3+1) (1+1)

⇒ d(1080) = 4 × 4 × 2

⇒ d(1080) = 32

Therefore, there are 32 divisors of 1080

Problem 6: Find the sum of divisors of 52

Solution:

It is calculated using the following formula:

σ(n) = 1 + n + ∑_{d|n, d≠1, d≠n} d

The divisors of 52 are 1, 2, 4, 13, 26, and 52

⇒ σ(52)=1+52+(2+4+13+26)

⇒ σ(52)=53+2+4+13+26

⇒ σ(52)=98

Therefore, the sum of divisors of 52 is 98

Problem 7: Find the φ(60) using Euler’s Totient Function formula

Solution:

It is calculated using the following formula:

ϕ(n) = n × (1 – 1/p₁) × (1 – 1/p₂) × . . . × (1 – 1/p_k)

The prime factorization of 60 is 2² x 3 x 5

⇒ ϕ(60) = 60 x ( 1 – 1/2) x (1 – 1/3) x (1 – 1/5)

⇒ ϕ(60) = 60 (1/2) (2/3) (4/5)

⇒ ϕ(60) = 24

So, φ(60) = 24. There are 24 positive integers less than or equal to 60 that are relatively prime to 60.

Divisor is the number from which we divide the dividend to determine the quotient and remainder. In arithmetic, division is one of the four fundamental operations; other operations are addition, subtraction, and multiplication.

Divisors in Number Theory are integers that divides another integer without leaving the remainder is also called a divisor.

In this article, we will discuss both definitions of a divisor, including the general, and the definition in number theory. We will also explore various properties and examples related to divisors and discuss concepts such as prime divisors, the number of divisors, the sum of divisors, and the difference between a divisor and a factor.