Factors of 256

As we all know that 256 is a composite number, it has some proper factors, i.e., the factors other than 1 and 256 itself. 256 has a total 9 of factors: $1,\text{ }2,\text{ }4,\text{ }8,\text{ }16,\text{ }32,\text{ }64,\text{ }128,\text{ }256$. Naturally, a prime factor of 256 is a factor of 256 which is also a prime number; hence from the above list of factors, we can see that 2 is the only prime factor of 256. Moreover, 256 is a perfect square, i.e., each prime factor of 256 occurs an even number of times. Here, the only prime factor of 256 is 2 and it occurs 8 times in the prime factorization of 256, i.e., $256=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2$. So, the square root of 256 is $2\times 2\times 2\times 2=16$. Also, the pair factors of 256 are (1,256), (2,128), (4,64), (8,32), and (16,16).

Definition of a Factor of an Integer

A factor of an integer is another integer that can divide the integer without leaving any remainder, i.e., an integer $F$ is said to be a factor of another integer $N$ if $F$ completely divides $N$. For example, the integers 2 and 3 can divide the integer 6 without leaving any remainder; so, 2 and 3 are factors of 6.

A factor can be both a positive and a negative integer that we will understand by the same example above. As 2 is a factor of 6, (-2) is also a factor of 6 because $6\div (-2)=-3$ which is also an integer, i.e., (-2) completely divides 6 without leaving any remainder. Similarly, (-3) is also a factor of 6. Thus, we can say that if $F$ is a factor of $N$, then $-F$ is also a factor of $N$. Here, in this article, we will mostly talk about the positive factors of 256.

Proper and Improper Factors

As 1 divides every integer and each integer divides itself, by definition of a factor, every integer $N$ has at least two factors: 1 and $N$ itself. These are called the Improper Factors of $N$. All other factors of $N$ are called the Proper Factors of $N$. For example, 1, 6 are the improper factors of 6 and $2,\text{ }3$ are proper factors of 6.

Prime Factors

As the name suggests, a prime factor of an integer $N$ is an integer $P$ that is a factor of $N$ as well as a prime number. For example, 2 and 3 are prime factors of 6 as they are factors of 6, and also, they are prime numbers.

Pair Factors

A pair of integers $\left( {{F}_{1}},{{F}_{2}} \right)$ are said to be a Pair Factors of an integer $N$, if firstly, they are factors of $N$ and secondly, their product yields $N$, i.e., ${{F}_{1}}\times {{F}_{2}}=N$. For example, as $1,\text{ }2,\text{ }3,\text{ }6$ are factors of 6 and $1\times 6=6$, $2\times 3=6$; $(1,6)$ and $(2,3)$ are pair factors of 6.

What are the Factors of 256?

Factors of 256 are the integers that completely divide it without leaving any remainder. Let us first find out the integers that fully divide 256:

$256\div 1=256$ $256\div 256=1$

$256\div 2=128$ $256\div 128=2$

$256\div 4=64$ $256\div 64=4$

$256\div 8=32$ $256\div 32=8$

$256\div 16=16$

Hence, by definition, 256 has a total of 9 factors: $1,\text{ }2,\text{ }4,\text{ }8,\text{ }16,\text{ }32,\text{ }64,\text{ }128,\text{ }256$.

If we consider negative factors also, then 256 has a total of 18 factors, viz., $\pm 1,\text{ }\pm 2,\text{ }\pm 4,\text{ }\pm 8,\text{ }\pm 16,\text{ }\pm 32,\text{ }\pm 64,\text{ }\pm 128,\text{ }\pm 256$.

Method of Finding the Factors of 256

In this section, we will learn how to find the factors of $256$. There are several methods of finding the factors of an integer. Here, we will follow the Division Method. In this method, we will divide $256$ successively by the integers starting from $1, 2, 3$ and so on and whenever we get the remainder $0$, then the corresponding divisor and quotient will be treated as the factors of $256$. This process continues till any one of the numbers is repeated.

In Step 16, the quotient and the divisor are the same, i.e., $16$. This means that all the factors of $256$ have been obtained. So, we must stop the process.

Hence, the factors of 256 are $1,\text{ }2,\text{ }4,\text{ }8,\text{ }16,\text{ }32,\text{ }64,\text{ }128,\text{ }256$. Moreover, notice that in Step-16, $256=16\times 16$, i.e., the number $16$ repeats itself as a factor. Hence, $256$ is a perfect square.

Factor Tree

Prime Factorization of $256$: Factor Tree

A prime number is a positive integer $P$ that has only two factors (not considering the negative factors here). For example, $2,\text{ }3,\text{ }5$ etc.

One of the most useful methods of finding the factors of an integer is prime factorization. In this method, we factorise an integer only into its prime factors. Let us find the prime factorization of $256$. This can be obtained by the following few steps.

1. In the first step, we take the smallest prime number $2$ and check whether it divides $256$ or not. Obviously, as $256$ is an even number, $2$ does divide 256 completely as it does not leave any non-zero remainder $(256=2\times 128+0)$ while dividing $256$. Therefore, $2$ is a prime factor of $256$.

2. Next, we take the quotient $128$ obtained in the first step and check whether $2$ divides $128$ or not. Clearly, $2$ divides $128$ and $128\div 2=64$.

3. Again, we take the quotient $64$ obtained in the second step and check whether $2$ divides $64$ or not. Clearly, $2$ divides $64$ and $64\div 2=32$.

4. Now, we proceed for the above quotient $32$ and the prime number $2$. Again, we can see that $32\div 2=16$; that means $2$ divides $32$.

5. Next, we take the above quotient $16$ and the same prime number $2$. Then, we can easily obtain that $16\div 2=8$.

6. Proceeding similarly, we will obtain: $8\div 2=4,\text{ }4\div 2=2,\text{ }2\div 2=1$.

As we have got $1$ as a quotient at the end, we cannot proceed further because $1$ is not a multiple of any prime numbers.

In brief, we perform the following in the above six steps:

$256\div 2 =128$

$128\div 2 =64$

$64\div 2 =32$

$32\div 2 =16$

$16\div 2 =8$

$8\div 2 =4$

$4\div 2 =2$

$2\div 2 =1$

Therefore, 2 is the only prime factor of $256$. Moreover, the prime factorization of $256$ is $256=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2$. This can be easily understood by the following factor tree of 256 and the upside-down division:

Factors of 256

Now, we can find all the factors of 256 by prime factorization as follows:

• First, write all the prime factors as many times as they occurred: $2,\text{ }2,\text{ }2,\text{ }2,\text{ }2,\text{ }2,\text{ }2,\text{ }2$.

• Now, multiply each of these number with another number and get: $2\times 2=4$., $2\times 2\times 2=8$, $2\times 2\times 2\times 2=16$, $2\times 2\times 2\times 2\times 2=32$, $2\times 2\times 2\times 2\times 2\times 2=64$, $2\times 2\times 2\times 2\times 2\times 2\times 2=128$, $2\times 2\times 2\times 2\times 2\times 2\times 2\times 2=256$. Thus, the factors obtained are: $2,\text{ }4,\text{ }8,\text{ }16,\text{ }32,\text{ }64,\text{ }128,\text{ }256$.

• Also, 1 is a factor.

• Now, list all the factors. Hence, all the factors of 256 are $1,\text{ }2,\text{ }4,\text{ }8,\text{ }16,\text{ }32,\text{ }64,\text{ }128,\text{ }256$.

Factors of 256 in Pairs

We have,

$256=1\times 256$

$256=2\times 128$

$256=4\times 64$

$256=8\times 32$

$256=16\times 16$

Hence, by definition, the pair factors of $256$ are $(1,256)$, $(2,128)$, $(4,64)$, $(8,32)$, and $(16,16)$.

Interesting Facts

• 256 is a perfect square. The square root of 256 is 16.

• The sum of all the factors of 256 is 511.

• Did you know that there are no factors of $N$ in between $\dfrac{N}{2}$ and $N$

Solved Examples

Example 1: There are 200 pens and 256 pencils to be distributed among the students of class VII. What should be the maximum number of students so that each student gets an equal number of pens and pencils?

Solution: Here, the maximum number of students should be the largest integer that divides both 200 and 256.

So, the required number of students will be the H.C.F. of 200 and 256.

By prime factorization, we get

$200=2\times 2\times 2\times 5\times 5$

$256=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2$

Hence, the Highest Common Factor of 200 and 256 is $2\times 2\times 2=8$.

Therefore, the maximum number of students should be 8.

Example 2: What is the sum of all factors of 256?

Solution: The factors of 256 are $1,\text{ }2,\text{ }4,\text{ }8,\text{ }16,\text{ }32,\text{ }64,\text{ }128,\text{ }256$. So, the sum of all factors of 256 is $1+2+4+8+16+32+64+128+256=511$.

Conclusion

As we have discussed above, rhere are several ways to find the factors of 256, including the division method and the prime factorization method. It has a total of 9 factors: $1,\text{ }2,\text{ }4,\text{ }8,\text{ }16,\text{ }32,\text{ }64,\text{ }128,\text{ }256$ among which, there is only one prime factor that is 2. Prime factorization of 256 is $256=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2$. 256 is a perfect square; the square root of 256 is 16.

Practice Questions

1. Find the largest factor of 256 which is a perfect square.

2. Find H.C.F. of 64 and 256.

3. Is there any factor of 256 which lies between 128 and 256? Justify your answer.

4. Find the average factor of 256. Is 40 a factor of 256?

Answer

1. 256

2. 64

3. No, because there are no factors of a number $N$ that lies in between $\dfrac{N}{2}$ and $N$.

4. No.