More FACTS about Factors In this article, we extend the application one very simple mathematical concept: factors. With the help of different examples, the article explores the different types of questions that can be based on factors, and how these should be approached. Example 1: Find the sum of divisors of 72. Solution:The first step that you carry out in this case is the prime factorization of 72 is = 23 * 32. Now in order to find the sum, firstly list down all the factors. Now the different ways of expressing powers of 2 and 3 are as follows: (20+21+22+23)*(30+31+32) = (1+21+22+23) (1+31+32) =1(1+3+32) +2(1+3+32) + 22(1+3+32)+23(1+3+32) =1+ 3 + 9 + 2 + 6 + 18 + 4 + 12 + 36 + 8 + 24 + 72 = Sum of factors of 72 Now algebraically it can be found as (24 – 1)*(33 – 1)/[(2-1)(3-1)]= 195. If you observe closely, the same formula is applied in the case of calculating the sum of a GP. It can be concluded that: Sum of factors = [(Pa+1-1)(qb+1-1)(rc+1-1)]/[(p-1)(q-1)(r-1)], where P, Q and R are the different prime factors of the number. Example 2: N=23*32*53. Find i) Number of Factors ii) Sum of factors Solution: i) The power of 2 can be selected in 4 ways. Similarly powers of 3 and 5 can be selected in 3 and 4 ways respectively. So any combination of 2’s, 3’s or 5’s power will give you the required factor. So, there are 4*3*4 = 48 factors. Solution: ii) Now the sum will be = (20+21+22+23)(30+31+32)(50+51+52+53) = [(24 -1)(33 -1)(54 -1)]/[(2-1)(3-1)(5-1)] = 30420. Example 3: How many factors of N= 22*33*51 are odd factors? Also, find their sum. Solution: For odd factors of N, the only primes worth consideration will be 33 and 51 as the power of 2 cannot be taken. Now 3 can be selected in 4 ways and 5 can be selected in 2 ways. Thus, the total number of odd factors is 4*2 = 8 ways. The sum of these will be (30+31+32+33) (50+51) = [(34 -1)(52 -1)]/[(3-1)(5-1)] = 240. Example 4: In how many ways 72 can be written as product of two factors? Solution: Let’s solve this question using the longer method first. The total factors of 72 can be listed as a combination of two factors in the following ways: 72 = 1*72 ; 2*36 ; 3*24 ; 4*18 ; 6*12; 8*9. So the number of ways is 6. Also, 72 = 23*32 has 4*3 = 12 factors. In this arrangement, all the 12 factors are written by taking two at a time. So, you get 12/2 = 6 ways. Example 5: Find product of factors of 72. Solution: Here the required product is: = (1*2*3*4*6*8*9*12*18*24*36*72) = (1*72)(2*36)(3*24)(4*18)(6*12)(8*9). It can be seen that every product is giving 72 and as per the last question 72 can be written as product of two factors in 6 ways. So answer will be 726 and it can be concluded that Product of factors of N is= (N)Total no. of factors/2
Posted on: Tue, 07 Oct 2014 05:15:14 +0000
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