Factors Of A Number: Definition, Properties, and Solved Examples

Factors of a number are the integers that divide the number exactly without leaving any remainder. Understanding the factors of a number is fundamental in mathematics as it lays the groundwork for more complex concepts like prime factorization, greatest common divisors, and least common multiples. The properties of factors include the fact that every number has at least two factors, 1 and itself, and that the factors of a number are always less than or equal to the number. In this article, we will delve into the definition and properties of factors, explore various methods to find them, and provide solved examples to solidify your understanding.

Definition of Factors of a Number?

Factors of a number are the integers that can be multiplied in pairs to produce that number. In other words, a factor of a number divides the number exactly without leaving any remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because each of these numbers divides 12 without a remainder.

Properties of Factors are stated below:

1. Every Number Has at Least Two Factors: Every number has at least two factors: 1 and the number itself.
2. Factors are Always Less Than or Equal to the Number: Factors of a number are always less than or equal to the number.
3. 1 is a Universal Factor: 1 is a factor of every number because every number divided by 1 is the number itself.
4. The Number is a Factor of Itself: Every number is a factor of itself because any number divided by itself is 1.
5. Factors Come in Pairs: Factors often come in pairs that, when multiplied together, produce the original number (e.g., for 12, the pairs are (1, 12), (2, 6), and (3, 4)).
6. Prime Numbers Have Only Two Factors: Prime numbers have exactly two distinct factors: 1 and the number itself.

Also Read: Questions of Logical Problems Reasoning

Properties of Factors of a Number with Formulas

Understanding the properties of factors of a number helps in simplifying mathematical problems and finding solutions efficiently. Here are the key properties along with their respective formulas:

Every Number Has at Least Two Factors

• Property: Every positive integer nnn has at least two factors: 1 and n itself.
• Formula: If n is a positive integer, then 1 and n are factors of n.

Factors are Always Less Than or Equal to the Number

• Property: The factors of a number n are always less than or equal to n.
• Formula: If d is a factor of n, then d ≤ n.

1 is a Universal Factor

• Property: 1 is a factor of every positive integer.
• Formula: For any positive integer n, 1 is a factor (n÷1 = n).

The Number is a Factor of Itself

• Property: Any number n is a factor of itself.
• Formula: For any positive integer n, n÷n=1.

Factors Come in Pairs

• Property: Factors of a number come in pairs that multiply to give the original number.
• Formula: If ddd is a factor of n, then there exists another factor n/d​ such that d×n/d=n.

Properties of Factors of a Number With Solved Examples

Here are five solved examples that demonstrate how to find and work with factors of a number:

Q1: Factors of 24

Solution:

1. Start by identifying the smallest factor pair: 1×24.
2. Continue finding pairs: 2×12, 3×8, 4×6.

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

Q2: Factors of 30

Solution:

1. Identify the smallest factor pair: 1×30.
2. Continue finding pairs: 2×15, 3×10, 5×6.

Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30

Q3: Factors of 50

Solution:

1. Identify the smallest factor pair: 1×50.
2. Continue finding pairs: 2×25, 5×10.

Factors of 50: 1, 2, 5, 10, 25, 50

Q4: Factors of 42

Solution:

1. Identify the smallest factor pair: 1×42.
2. Continue finding pairs: 2×21, 3×14, 6×7.

Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42

Q5: Factors of 100

Solution:

1. Identify the smallest factor pair: 1×100.
2. Continue finding pairs: 2×50, 4×25, 5×20, 10×10.

Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100

Also Read: What are Composite Numbers from 1 to 100?

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