Factors of 76

The factors of any particular number are generally defined as the natural numbers that when multiplied will give the original number, in other words when multiplying the two factors we obtain the result or the product of those numbers as the original number. The factors of any natural number could be positive integers or negative integers i.e., the sign of the factors can be either positive or negative. Factors of 76 are all the integers that are capable of dividing the given number evenly, so here our aim is to find the natural numbers that can divide the given number 76 evenly.

From the definition of the factors, we know that the factors of a number are nothing but the product of certain numbers which completely divide the number under consideration. As previously mentioned the factors of any number can be either positive or negative integers. By multiplying the factors of a number we get the given number.

Let us understand the definition with an example, 1, 2, 3, 4, 6 and 12 are the factors of 12. On multiplying two or more numbers we obtain 12. Hence we have $4\times3=12~ or~ 1\times12=12$. In this article, we will study the factors of 76, how to find the factors of 76 and examples.

What are the Factors of 76?

Factors of a number is an important subtopic from number systems. Each prime number will have only two factors, i.e. 1 and number itself, whereas each composite number will have more than two factors, that include prime factors also. So for factors of 76, we can say all the numbers which completely divides 76 are the factors of 76. For example 1, 2, 4 etc.

How to Find the Factors of 76?

According to the definition of the factors, we know that factors are the numbers that divide the given number evenly and they can be either positive or negative integers.

Factors of a number can be determined by using various methods. One of the basic and easy methods is trial division. Using trial division, factors of the number 76 can be determined as follows:

• First of all, find the square root of 76 which is approximately 8.71, and now round it off to the closest integer which is 9.

• Start with 1 now, so the corresponding factor pair is 76 ÷ 1 = 76. So 1 and 76 are factor pairs.

• Do the same with the number 2 and proceed to test all integers (n ÷ 2, n ÷ 3, n ÷ 4... n ÷ s) up to the square root rounded to s. You need to record the factor pairs where division results in whole integer numbers with no remainders. Divide the number 76 by 2.

$76\div 2=38$

• Now if we repeat it with 3 then we won’t get the corresponding factor pair as a whole number, So it won’t be a factor of 76.

• Now repeat it with the next whole number which is 4, so the corresponding factor pair is $76\div 4=19$. So 4 and 19 are factor pairs.

• Similarly, we can check for the next numbers but 76 is not divisible by any other number which is less than or equal to 9.

Therefore, the factors of 76 are 1, 2, 4, 19, 38 and 76.

In simpler words, these are the numbers that divide the number 76 completely or the remainder will be zero. We know that the factors of a number can be either positive or negative numbers, hence the negative factors of 76 are $-1, -2, -4, -19, -38~ and -76$.

Factors of 76 can be Listed as Below:

• Positive factors of 76 = 1,2, 4, 19, 38 and 76.

• Negative factors of 76 = -1,-2, -4, -19, -38 and -76.

Hence we have in total 6 positive factors of 76 and 6 negative factors of 76.

Pair Factors of 76

Pair factors of 76 are the combination of two numbers which on multiplication gives the product as 76. In other words, the pair factors of 76 are the combination of two factors whose product will be equal to 76. Let us now discuss the pair factors of 76.

From the factors of 76 we have the following combinations whose product is 76:

$\Rightarrow 1\times 76=76$

$\Rightarrow 4\times 19=76$

$\Rightarrow 2\times 38=76$

$\Rightarrow (-1)\times( -76)=76$

$\Rightarrow( -4)\times (-19)=76$

$\Rightarrow (-2)\times (-38)=76

Therefore pair factors of 76 are $(1\times 76), (2\times38),(4\times19)(-1\times (-76)), (-2\times(-38)),(-4\times(-19))$. Here we have considered negative pair factors as well but generally we take only positive factors.

Prime Factorization of 76

The prime factors are defined as the product of all the factors that are prime numbers. We know that prime numbers are natural numbers that can be divided by themselves and one. So now we can find the prime factors of 76 from the list of factors of 76.

Let us find the prime factors of 76 by expressing the number 76 as a product of prime numbers. From prime factorization, we find that 76 can be written as the product of $2\times2\times19=2^2\times19$. Thus it is clear from prime factorization that the prime factors of 76 are 2 and 19.

Thus the prime factors of 76 are 2 and 19.

Did You Know

• 76 is one of the two-digit natural numbers whose square and higher powers result in a number that concludes with 76.

$\Rightarrow76^2=5776$

$\Rightarrow76^3=438976$

$\Rightarrow76^4=33362176$

• 1 and the considered number itself is the factor of any chosen number.

• The numbers that contain more than 2 factors are known as composite numbers.

• 1 is not a prime number; 2 is the smallest possible prime number.

Solved Examples

1. What are the factors of 76, write down the prime factors of 76.

Sol: The factors of 76 are as listed below:

$\Rightarrow76\div2=38$

$\Rightarrow76\div4=19$

Also,

$\Rightarrow76\div1=76$

Therefore, the factors of 76 are 1, 2, 4, 19, 38 and 76.

Now, the prime factors of 76 are the factors of 76 that are prime numbers. From the prime factorization, we can find the prime factors of 76.

$\Rightarrow2\times2\times19=2^2\times19$

Therefore the prime factors of 76 are 2 and 19.

2. What are the factors of 90, write down the pair factors of 90.

Sol: The factors of 90 are as listed below:

$\Rightarrow90\div1=90$

$\Rightarrow90\div2=45$

$\Rightarrow90\div3=30$

$\Rightarrow90\div5=18$

$\Rightarrow90\div6=15$

$\Rightarrow90\div9=10$

Therefore, the factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90.

Now, the pair factors of 90 are

$(90\times1), (45\times2), (30\times3), (18\times5) (15\times6), (10\times9)$.

Practice Questions

1. Find the factors of 32?

2. Find the factors of 84?

Conclusion

In this article we have learned about Factors and how to factorize a number. Then we studied the factors and factor pairs of 76. We have studied the negative factors as well but generally we don’t consider them.