Factors of 77

Before we start with the what are the factors of 77, let us first discuss the definition of factors in mathematics. Basically, a factor is a natural number that results in the null remainder on division or in other words factor is a natural number that divides the chosen number completely. The factors of a number can be either a positive integer or a negative integer i.e., the sign of the factors can be either plus or minus.

Let us understand the definition with a suitable example, let's check for the factors of 10. Since 10 can be factorized as $1\times10~and~2\times5$ and we know that the product of two negative numbers is a positive number only. Therefore, the factors are 10 are actually $1, -1, 2, -2, 5, -5, 10~and~ -10$. But when it comes to examples corresponding to the factors, generally positive numbers are considered, that to a whole number and a non-fractional digits.

Properties of Factors

Factors of natural numbers have certain properties. The characteristics of the factors are the following:

• The number of factors is finite. The factor is always less than or equal to the specified number.

• All numbers except 0 and 1 have at least two factors, 1 and itself.

• Division and multiplication are operations used to find factors.

What are the Factors of 77?

In mathematics, a factor is a number that equally divides another number. The factor can also be an algebraic expression that also divides another expression.

In other words we can say that factors of a number are the numbers which when multiplied together then result in the original number. For example, $(2\times4)=8$ so we can say 2 and 4 are the factors of 8.

So for factors of 77, we can say all the numbers which completely divide 77 are the factors of 77. For example 1, 7, 11 etc.

How to Find the Factors of 77

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 77 can be determined as follows:

• First of all, find the square root of 77 which is approximately 8.77, and now round it off to the closest integer which is 9.

• Start with 1 now, so the corresponding factor pair is $77\div1=77$. So 1 and 77 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.

• Now if we repeat it with 2 then we won’t get the corresponding factor pair as a whole number, So it won’t be a factor of 77.

• 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 77.

• Now if we repeat it with 4 then we won’t get the corresponding factor pair as a whole number, So it won’t be a factor of 77.

• Now if we repeat it with 5 then we won’t get the corresponding factor pair as a whole number, So it won’t be a factor of 77.

• Now if we repeat it with 6 then we won’t get the corresponding factor pair as a whole number, So it won’t be a factor of 77.

• Now repeat it with the next whole number which is 7, so the corresponding factor pair is 77 ÷ 7 = 11. So 7 and 11 are factor pairs.

• Similarly, we can check for the next numbers but 77 is not divisible by any other number which is less than or equal to 9.

Therefore, the factors of 77 are $1, 7, 11~and~77$.

In simpler words, these are the numbers that divide the number 77 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 77 are $-1, -7, -11~and~-77$

Factors of 77 can be Listed as Below:

• Positive factors of 77 = 1, 7, 11 and 77.

• Negative factors of 77 = -1,-7, -11 and -77.

Hence we have in total 4 positive factors of 77 and 4 negative factors of 77.

Pair Factors of 77

Pair factors of 77 are the combination of two which on multiplication gives the product as 77. In other words, the pair factors of 77 are the combination of two factors whose product will be equal to 77. Let us now discuss the pair factors of 77.

From the factors of 77 we have the following combinations whose product is 77:

$\Rightarrow1\times77=77$

$\Rightarrow7\times11=77$

$\Rightarrow(-1)\times(-77)=77$

$\Rightarrow(-7)\times(-11)=77$

Therefore Pair Factors of 77 are $(1\times77),~(7\times11),~((-1)\times(-77)),~((-7)\times(-11))$.

Prime Factorization of 77

The prime factors are defined as the factors that are prime numbers. We know that prime numbers are the natural number that can be divided by themselves and one. So now we can find the prime factors of 77 from the list of factors of 77.

Let us find the prime factors of 77 by expressing the number 77 as a product of prime numbers. From prime factorisation, we find that 77 can be written as the product of $7\times11=77$. Thus it is clear from prime factorization that the prime factors of 77 are 7 and 11.

Thus the prime factors of 77 are 7 and 11.

Did You Know?

The natural number 77 is actually a unique composite number which is a palindrome and can be represented as a product of two prime numbers 7 and 11.

Solved Examples

1. What are the factors of 8, write down the prime factors of 8.

Sol: The factors of 8 are as listed below:

$\Rightarrow8\div2=4$

Also,

$\Rightarrow8\div1=8$

Therefore, the factors of 8 are , 2, 4 and 8

Now, the prime factors of 8 are the factors of 8 that are prime numbers. From the prime factorization, we can find the prime factors of 8.

$\Rightarrow2\times4=2\times2^2=2^3=8$

Therefore the prime factor of 8 is only 2.

2. What are the factors of 15, write down the pair factors of 15.

Sol: The factors of 15 are as listed below:

$\Rightarrow15\div3=5$

And,

$\Rightarrow15\div1=15$

Therefore, the factors of 15 are 1, 3, 5 and 15.

Now, the pair factors of 15 are $(3\times5)~and~(1\times15)$.

Practice Questions

1. Find the factors of 42?

2. Find the factors of 39?

Conclusion

In this article we have learned about Factors and properties of factors. Then we studied the factors and factor pairs of 77 and how to find factors of 77. We have studied the negative factors as well but generally we don’t consider them.