# Factors of 35: Prime Factorization, Methods, Tree, and Examples

**Factors of 35 **are the numbers that evenly divide 35 without leaving any remainder. Factors are always in the form of whole numbers.

**Factoring** is a **mathematical technique** used to solve many algebraic equations. When we multiply two different numbers to get a specific product. The multiplied numbers are called factors of that product.

There are two types of factors:

- Positive Factors
- Negative Factors

In mathematics, there are two ways to find factors of a number. One is the multiplication method the other one is the division method.

There are many real-life examples related to factors. For example, dividing sweets among children, arranging biscuits in boxes, distributing pencils among students, etc.

In this article, we will learn about factors of 35, methods to find them,** factor tree**, examples, and much more.

## What Are the Factors of 35?

**Factors of 35 are 1, 5, 7, and 35; all of these numbers divide 35 evenly. The remainder is zero.**

35 is an **odd composite number**. A number having more than two factors is known as the composite number. There are eight total factors of 35. Four are positive factors and the other four are negative factors.

## How To Calculate the Factors of 35?

You can calculate the **factors of 35** by two methods: division and multiplication method. We will discuss both ways in this article.

As the number 35 is composite, there are more than two factors of 35. Make a number line starting from 1 and ending on 35. We have to find the factors in between them.

Factors of 35 by division method:

**One is a factor of every whole number** because every number is completely divided by 1.

\[ \frac{35}{1} = 35 \]

\[ \frac{35}{-1} = -35 \]

**1 and -1 are factors of 35.**

35 is not even, so it wouldn’t be divided by 2.

Let’s divide 35 by 3:

\[ \frac{35}{3} = 11.66 \]

When we divide 35 by 3, the number is not evenly divided. The remainder is 2. Condition of factors is not satisfied 3 is not a factor of 35.

Divide 35 by 5:

\[ \frac{35}{5} = 7 \]

\[ \frac{35}{-5} = -7 \]

When 35 is divided by 5. The number is not evenly divided. The remainder is 0. Condition of factors is satisfied** 5 and -5 are the factors of 35.**

Divide 35 by 6:

\[ \frac{35}{6} = 5.83 \]

When we divide 35 by 5, the condition of factors is not satisfied. The remainder is 5. As a result of the above calculation 6 is not a factor of 35.

Divide 35 by 7:

\[ \frac{35}{7} = 5 \]

\[ \frac{35}{-7} = -5 \]

When 35 is divided by 7. The remainder is 0. Condition of factors is satisfied** 7 and -7 are the factors of 35.**

Divide 35 by 11:

\[ \frac{35}{11} = 3.18 \]

When 35 is divided by 11. Condition of factors is not satisfied. The remainder is 2. As a result of the above calculation 11 is not a factor of 35.

Every number is a factor in itself. As every number divides itself evenly and the remainder is always zero. **35 and -35 are the factors of 35**.

**Positive factors of 35 = 1, 5, 7, 35. **

**Negative factors of 35 = -1, -5, -7, -35. **

Factors of 35 by multiplication method:

**1 x 35 = 35 **

**-1 x -35 = 35 **

When a negative sign is multiplied by a negative sign, the product is always positive.

By the above multiplication, we conclude that 1, -1, 35, and -35 both are factors of 35

**5 x 7 = 35 **

**-5 x -7 = 35 **

**Factors of 35 are 1, -1, 5, -5, 35, and -35.**

## Factors of 35 by Prime Factorization

The technique used to write the number 35 as the product of its prime factors is known as **Prime Factorization**.

**Prime factorization** is a mathematical process in which we discover the prime factors of a number, and we obtain the original number when multiplied together. This method is only applicable to composite numbers.

The two most common ways of finding prime factorizations are the following:

- Division method.
- Factor tree.

Finding prime factorization by division method:

Firstly, divide the number 35 by the smallest prime factor. The smallest prime factor in the list of factors of 35 is 5.

which is 5.

\[ \frac{35}{5} = 7 \]

7 is the quotient. It is not divisible by 5; divide it by the next prime factor. The next smallest prime factor is 7.

\[ \frac{7}{7} = 1 \]

The quotient is 1, so this division ends here.

The **Prime Factorization of 35** is shown below in figure 1:

**The Highest Common Factor** of two integers is the greatest number from the list of factors of both numbers that divides both numbers evenly, and the remainder is zero. The highest common factor between 35 and 70 is 35.

**The Least Common Factor** of two integers is the smallest number from the list of factors of both numbers that divides both numbers evenly, and the remainder is zero. The least common factor between 35 and 70 is 5.

## Factor Tree of 35

The **factor tree** is a pictorial representation of factors of a number, specifically the prime factors. A factor tree is just like a tree that has many branches. Every branch further splits with some logic.

Now we will learn how to construct a factor tree:

Write the number at the top. Draw two branches out of it. Fill these branches with the factors of the number. Keep on dividing until each branch ends up with the prime factors.

The **factor tree of 35** is shown below in figure 2:

The prime factorization of 35 can be written as:

**Prime Factorization of 35 : 5 x 7 **

## Factors of 35 in Pairs

Writing a set of two **factors of 35. **When multiplied gives a particular answer, which is equal to the original number.

Factor pairs of a number can be calculated by the simple multiplication method. Factor pairs can be positive and negative, but they can’t be in fractional form.

Finding **factor pairs **using the multiplication method:

**1 x 35 = 35 **

**5 x 7 = 35 **

The** positive factor pairs of 35** are the following:

**(1, 35)**

**(5, 7)**

Finding **negative factors of 35**:

**-1 x -35 = 35 **

**-5 x -7 = 35 **

The **negative factor pairs of 35** are the following:

**(-1, -35)**

**(-5, -7)**

## Factors of 35 Solved Examples

Following are some solved examples for a better understanding regarding factors of 35.

### Example 1

Rachel has 35 red boxes and Maya has 75 green boxes. They want to arrange the boxes in such a way that each row contains an equal number of boxes and also each row should have only red boxes or green boxes. What is the greatest number of boxes that can be arranged in each row?

### Solution

The given condition is:

The number of boxes should be equal in each row.

Each row should have a single color of boxes.

To arrange the green and red boxes in an equal number of rows find the **largest common factor** between 35 and 75.

Firstly, find the factors of numbers 35 and 75 are as follows:

**Factors of 35 = 1, 5, 7, 35 **

**Factors of 75 = 1, 3, 5, 15, 25, 75**

From the list of **factors of 35** and **75**. Now find the HCF (Highest Common Factor).

**GCF of 35 and 75 = 5 **

5 is also a common factor of 35 and 75.

**Each row will have 5 boxes **

Rows of red boxes: \[ \frac{35}{5} = 7 \]

Rows of red boxes: \[ \frac{75}{5} = 15 \]

### Example 2

Find the sum of all factors of 35 and divide it by the sum of even factors of 35.

### Solution

Factors of 35 = 1, 5, 7, 35.

Finding the sum of all factors of 35

**Sum: 1 + 5 + 7 + 35 = 48 **

35 is an odd number, and factors of 35 are also odd.

\[ \frac{48}{1} = 48 \]

### Example 3

Bela has 15 pineapples, 25 apricots, and 35 pears. She wants to place all fruits into baskets, with each basket having the same number of pieces of fruit in it. Without mixing the fruit, what is the greatest number of pieces of fruit that are placed in each basket?

### Solution

Fruits Bela has:

The number of pineapples: 15

The number of apricots: 25

The number of pears: 35

To find the greatest/highest common factor. First, we have to calculate the factors for 15, 25, and 35.

**Factors of 15 = 1, 3, 5, 15 **

**Factors of 25 = 1, 5, 25 **

**Factors of 35 = 1, 5, 7, 35 **

The highest common factor of 15, 25, and 35 is 5.

There will be 5 baskets.

Now divide the fruits into the baskets.

The number of pineapples in each basket: \[ \frac{15}{5} = 3 \]

The number of apricots in each basket: \[ \frac{25}{5} = 5 \]

The number of pears in each basket: \[ \frac{35}{5} = 7 \]

Each basket contains 3 pineapples, 5 apricots, and 7 pears.

*Images/mathematical drawings are created with GeoGebra. *