Question

What are the divisors of 35?

Short answer

The divisors of 35 are 1, 5, 7, and 35.

Step-by-step solution

Understand What a Divisor Is

A divisor of a number is any integer that divides that number without leaving a remainder. For example, if a number 'd' divides another number 'n' perfectly, then 'd' is a divisor of 'n'.

Test Small Integers

Start testing small integers on the number 35 to see which ones divide it without leaving a remainder. For example, check if 1, 2, 3, 4, and so on, divide 35.

Check Division by 1 and 35

Every number is divisible by 1 and itself. Therefore, 1 and 35 are divisors of 35.

Check Other Possible Divisors

Next, test other possible numbers. 35 is not divisible by 2, 3, or 4 as they leave a remainder. However, dividing 35 by 5 results in 7 with no remainder. Thus, both 5 and 7 are divisors. Similarly, 7 divides 35 without leaving a remainder.

List All Divisors

Based on the steps above, we've found all the divisors of 35: 1, 5, 7, and 35.

Key concepts

prime factorization

Prime factorization is a key concept in understanding the divisors of a number. It involves breaking down a number into its prime factors, which are prime numbers that multiply together to give the original number. For example, the prime factorization of 35 is 5 and 7 because 5 × 7 = 35. Knowing the prime factors helps in identifying all possible divisors of the number. In our case, since 35 can be expressed as the product of its prime factors (5 and 7), we can easily determine that 1, 5, 7, and 35 are its divisors. The process involves:

• Finding the smallest prime number that divides the target number
• Dividing the target number by this smallest prime number
• Repeating the process on the quotient until you reach 1

This systematic approach ensures that you don’t miss any divisors and helps in simplifying complex numbers to understand their basic building blocks.

number theory

Number theory is a branch of mathematics that deals with the properties and relationships of numbers, especially integers. It includes various concepts such as divisibility, prime numbers, and the Greatest Common Divisor (GCD). Understanding these ideas can help you solve problems related to divisors more effectively.
One key aspect to grasp is the concept of divisibility. A number 'a' is said to be divisible by another number 'b' if dividing 'a' by 'b' leaves no remainder. For example, 35 is divisible by 5 and 7 because 35 ÷ 5 = 7 and 35 ÷ 7 = 5.
Number theory also explores properties of numbers such as:

• Prime numbers and composite numbers
• Even and odd numbers
• Multiples and factors

These fundamentals are essential for understanding how numbers interact with each other and for solving more complex mathematical problems.

basic arithmetic operations

Basic arithmetic operations form the foundation of mathematics and are essential when dealing with divisors of a number. These operations include addition, subtraction, multiplication, and division.
When finding the divisors of a number like 35, you use multiplication and division extensively. For example, to check if a number like 5 is a divisor of 35, you divide 35 by 5. If the result is an integer with no remainder, then 5 is a divisor.
Here’s how basic arithmetic operations help in understanding divisors:

• Addition and Subtraction: Used less frequently in finding divisors but are fundamental for understanding the relationships between numbers.
• Multiplication: Critical for finding product-based relationships, like prime factorization (e.g., 5 × 7 = 35).
• Division: Essential for testing if one number is a divisor of another. For instance, 35 ÷ 5 = 7 confirms that 5 is a divisor.

Mastering these basic operations allows you to approach more complex mathematical concepts with confidence and ease.