Question

Find the number of factors of 512.
A) 8
B) 10
C) 4
D) 14

Answer 1

Hint: In this question, convert the given number N into the product of prime factors by prime factorization method. Then we will write the result of prime factorization in the form of N = ${M^a} \times {N^b} \times {O^c}$where M, N, and O are the prime numbers and a, b and c are their respective powers. Finally, we will use the following formula to find the total number of factors.
Total number of factors = (a +1) (b + 1) (c + 1)

Complete step-by-step answer:
Prime factorize 512
$512 = 2 \times 256$
$512 = 2 \times 2 \times 128$
$512 = 2 \times 2 \times 2 \times 64$
$512 = 2 \times 2 \times 2 \times 2 \times 32$
$512 = 2 \times 2 \times 2 \times 2 \times 2 \times 16$
$512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 8$
$512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 4$
$512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = {2^9}$
Number of factors = (9 + 1) = 10
Therefore, the number of factors of 512 is 10.

Note: Factors are the number which is multiplied to get another number. Prime factorization is a method of factoring a number in terms of prime numbers that is the factors will be prime numbers. Prime factorization is a way of finding the prime factors of a number that is prime numbers which can be multiplied together to get the original number. Prime factorization of 24 is $2 \times 2 \times 2 \times 3$. When prime numbers when multiplied by any natural numbers or whole numbers, gives composite numbers. Prime factors can divide the number completely. There are two types of prime factorization methods: 1) Division Method 2) Factor Tree method.
Division Method – In this method we will divide the given number by the smallest prime number. Again, we will divide the quotient by the smallest prime number. We will repeat the process until the quotient becomes 1. Finally, we will multiply all the prime factors.
Factor Tree method – We have already used this method to get the prime factors of 512.