Find the sum of all the factors of 78. (a) 168 (b) 170 (c) 167 (d) 189
Hint: First, start by defining a factor of a number. Then, perform the prime factorization of 78. Next, find all the factors of 78 using this prime factorization using the fact that the factors of 78 are formed by taking these prime factors individually and then in groups. Find the sum of all these factors which is your final answer.
Complete step-by-step solution - In this question, we need to find the sum of all the factors of the number 78. Let us first see the definition of a factor. In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder. Let us now find the factors of 78. For this, we will prime factorise 78: 2 |78 3 |39 13|13 |1 So, the prime factorization of 78 is 2 $\times $ 3 $\times $ 13 Now, the factors of 78 are formed by taking these prime factors individually and then in groups. Doing that, we get the following: The factors of 78 are: 1, 2, 3, 6, 13, 26, 39 and 78. Required sum = 1 + 2 + 3 + 6 + 13 + 26 + 39 + 78 = 168. Hence, option (a) is correct.
Note: In this question, it is very important to know what a factor of a number is. Also, it is important to know how to perform prime factorization of a number and then how to use this prime factorization to get all the factors of the number. Do not forget to consider 1 as a factor.