Factors of 34 | Prime Factorization of 34 - Explained Simply

Today we are going to present here Factor Tree of 34. The factor is the number that divides the original number. The Factors of 34 are 1, 2, 17 and 34 itself.

What are the factors of 34?

To find the factors of 34, we look for all integers that divide 34 evenly, meaning without a remainder. We can systematically check integers starting from 1.

Factor Tree Method of 34: Explained Simply

There are a couple of straightforward methods to find all the factors of a number like 34. We'll explore both.

Division Test Method of 34

This method involves testing numbers, starting from 1, to see if they divide 34 evenly.

Start with 1:

34 ÷ 1 = 34 (remainder 0)

This tells us 1 and 34 are factors of 34.

Try 2:

34 ÷ 2 = 17 (remainder 0)

This tells us 2 and 17 are factors of 34.

Try 3:

34 ÷ 3 = 11 with a remainder of 1.

So, 3 is not a factor of 34.

Try 4:

34 ÷ 4 = 8 with a remainder of 2.

So, 4 is not a factor of 34.

Try 5:

34 ÷ 5 = 6 with a remainder of 4.

So, 5 is not a factor of 34.

Why stop here? We don't need to check numbers greater than the square root of 34 (which is approximately 5.8). If we haven't found a factor by this point, its "partner" factor would have already been identified. For example, when we divided by 2 and got 17, we already found a factor (17) that is greater than 5. We've effectively found all pairs.

From this division process, we have identified the factors: 1, 2, 17, 34.

Finding Multiplication Pairs of 34

This method involves finding pairs of numbers that, when multiplied together, give you the target number (34 in this case).

Pair 1: What times what equals 34? Start with 1.

1 × 34 = 34

(Factors found: 1, 34)

Pair 2: Move to the next whole number, 2.

2 × 17 = 34

(Factors found: 2, 17)

Continue Testing (Mentally or on Paper):

Can we multiply 3 by something to get 34? No (3 × 11 = 33, 3 × 12 = 36).

Can we multiply 4 by something to get 34? No (4 × 8 = 32, 4 × 9 = 36).

Can we multiply 5 by something to get 34? No (5 × 6 = 30, 5 × 7 = 35).

Notice that as we test numbers like 3, 4, 5, their "partners" would be greater than 17. Since we've already found 17 (from 2 × 17), we know we've covered all possibilities. We stop when the numbers we're testing start to repeat or go past the square root of the number.

By combining the numbers from our multiplication pairs, we get the complete list of factors.

The factors of 34 are: 1, 2, 17, 34.

Prime Factorization of 34

Now, let's apply these methods to our target number: 34.

Using the Factor Tree for 34:

Start with 34.

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Find two factors of 34. We can see that 34 is an even number, so it's divisible by 2.

34 ÷ 2 = 17.

So, our factors are 2 and 17.

```

/ \

2 17

```

Check if these factors are prime:

2 is a prime number (it's only divisible by 1 and 2). Circle it.

17 is a prime number (it's only divisible by 1 and 17). Circle it.

```

/ \

② ⑰

```

All branches end in prime numbers.

The prime factorization of 34 is the product of the circled numbers: 2 × 17.

Using the Division Method for 34:

Write 34.

```

| 34

```

Divide 34 by the smallest prime number that divides it evenly. Since 34 is even, we start with 2.

34 ÷ 2 = 17.

```

2 | 34

| 17

```

Now, consider the new quotient, 17. What is the smallest prime number that divides 17 evenly?

17 is a prime number itself, so the only prime that divides it is 17.

17 ÷ 17 = 1.

```

2 | 34

17| 17

| 1

```

The quotient is 1, so we stop.

The prime factorization of 34 is the product of the prime numbers on the left side: 2 × 17.

Both methods confirm that the prime factorization of 34 is 2 × 17.