Factors are numbers that can be multiplied together to produce a given number. In the case of 34, its factors are the numbers that divide evenly into it without leaving a remainder. The factors of 34 are 1, 2, 17, and 34. These factors demonstrate that 34 is a composite number, as it has factors other than 1 and itself.

Factor 1 is always present for any number, while 2 and 17 are prime factors of 34. Prime factors are numbers that are only divisible by 1 and themselves, and in the case of 34, 2, and 17 fulfill this condition. Finally, the number 34 itself is also a factor, as any number is divisible by itself.

Understanding the Factors of the number 34 is useful for various mathematical operations and problem-solving. For instance, finding the prime Factors of the number 34 helps in simplifying fractions and determining the greatest common divisor between 34 and another number.

Additionally, knowing the Factors of the number 34 can aid in identifying patterns and relationships within a set of numbers or when exploring divisibility rules. By examining the factors, mathematicians and students can gain insights into the properties and behavior of this particular number.

**What are the factors of 34?**

Factors are numbers that can be multiplied together to produce a given number. In the case of 34, its factors are the numbers that divide evenly into it without leaving a remainder. The Factors of the number 34 are 1, 2, 17, and 34. These factors demonstrate that 34 is a composite number, as it has factors other than 1 and itself.

**Factors of 34:**1, 2, 17 and 34**Negative Factors of 34:**-1, -2, -17 and -34**Prime Factors of 34:**2, 17**Prime Factorization of 34:**2 × 17 = 2 × 17**Sum of Factors of 34:**54

**Key points:**

- A factor of a number is a divisor that evenly divides the number without leaving a remainder. In the case of the number 34, its factors include 1, 2, 17, and 34.
- Factor 1 is common to all numbers, including 34.
- Each number is a factor of itself, so 34 is also a factor of 34.
- The Factors of the number 34 are all less than or equal to 34, as every factor must be smaller than or equal to the original number.
- The factors of a given number are finite, including the Factors of the number 34.
- Among the Factors of the number 34, the pair consisting of 2 and 17 is an example of a pair of co-prime numbers, meaning they have no common factors other than 1.
- The highest common factor (H.C.F.) of two or more numbers is the largest factor that divides all the given numbers without leaving a remainder.

**Co-Prime Numbers as Factors of 34**

To determine the co-prime Factors of the number 34, we need to find the prime factors 34 and identify which of them are co-prime.

The prime factorization of 34 is as follows:

34 = 2 * 17

The prime factors of 34 are 2 and 17. To determine if they are co-prime, we need to check if they have any common factors other than 1.

In this case, 2 and 17 do not have any common factors other than 1. Therefore, they are co-prime factors 34.

So, the co-prime factors of 34 are 2 and 17.

**How to Find Factors of 34?**

To find the Factors of the number 34, you need to determine the numbers that divide evenly into 34 without leaving a remainder. Here’s how you can do it:

- Start by listing the first few positive integers: 1, 2, 3, 4, 5, 6, and so on.
- Check if each number divides evenly into 34. You can do this by performing the division and checking if the remainder is 0.
- If the remainder is 0, then the number is a factor of 34. Write it down.
- Continue this process until you reach the halfway point of 34. Since factors occur in pairs, you only need to check numbers up to the square root of 34 (approximately 5.83).
- In this case, the Factors of the number 34 are 1, 2, 17, and 34.

So, the factors are 1, 2, 17, and 34.

**Pair Factors of 34**

Let’s practice finding the factors of the number 34.

First, write down the number that needs to be factored in. In this case, it is 34.

Find the two numbers whose product equals 34 in step 2.

Positive Pair Factors of 34 | Negative Pair Factors of 34 |

1×34 | -1x-34 |

2×17 | -2x-17 |

**Factors of 34 by Division Method:**

Division | Factor | Remainder |

34 ÷ 2 | 2 | 1 |

34 ÷ 17 | 17 | 1 |

**Prime Factorization of 34**

To find the prime factorization of 34, you need to express it as a product of its prime factors. Here’s how you can do it:

- Start by dividing 34 by the smallest prime number, which is 2. If 2 divides evenly into 34, write down 2 as a factor and divide 34 by 2. 34 ÷ 2 = 17

Since 2 does not divide evenly into 34, we move on to the next prime number.

- The next prime number after 2 is 3. Check if 3 divides evenly into 34. In this case, it does not divide evenly.
- Continue with the next prime number, which is 5. Again, check if 5 divides evenly into 34. It does not.
- Next, try dividing 34 by 7. It also does not divide evenly.
- Finally, try dividing 34 by 17. It divides evenly into 34. 34 ÷ 17 = 2

Since 17 is a prime number and it divides evenly into 34, we can stop here.

The prime factorization of 34 is therefore:

So, the prime Factors of the number 34 are 2 and 17.

**Factor Tree of 34:**

To find the factor tree of 34, we need to break it down into its prime factors. Here’s how you can do it:

**Step 1:** Start with the number you want to factor, which is 34.

**Step 2:** Look for a pair of factors whose product equals 34. The factors should be prime numbers, so let’s start with the smallest prime number, 2. We can see that 2 is a factor of 34 because 2 divides 34 evenly (34 ÷ 2 = 17).

**Step 3:** Now, we need to find the factors of 17. Since 17 is a prime number, we can’t break it down any further. Therefore, the factor tree is complete:

This shows that the prime Factors of the number 34 are 2 and 17.

**Example 1:**

Find the HCF of 34 and 51.

Solution: The HCF of 34 and 51 is 17.

**Example 2:**

Find the LCM of 34 and 51.

Solution: The LCM of 34 and 51 is 102.

**Example 3:**

Find the common factors of 51 and 34.

Solution: The common factors of 34 and 51 are 1 and 17.

**Example 4:** Find the HCF of 68 and 34.

Solution: The HCF of 34 and 68 is 34.

**Example 5:**

Find the LCM of 68 and 34.

Solution: The LCM of 34 and 68 is 68.

**Are 3 and 6 factors of 34?**

Yes, both 3 and 6 are 34’s factors.

**Find a factor of 34 that is a multiple of 3.**

The factor of 34 which is a multiple of 3 is 6.

**What is the sum of the Factors of the number 34?**

The sum of the Factors of the number 34 is 1 + 2 + 17 + 34 = 54.

**Is 34 a perfect square? Why or why not?**

No, 34 is not a perfect square because it cannot be expressed as the square of an integer.