Quant pill for the week: Answers

The first thing which we need to do is to factorize 2268 with the help of divisibility rules :

2268 = 2²  × 3⁴ × 7¹

1. The number of ways of writing a number as a product of two co-prime numbers = 2^n-1  where n= the number of prime factors of a number. Since there are only 3 prime factors that is 2, 3 and 7. Hence, there are 2^3-1 = 4 ways.

2. Let’s take a smaller number say 24. If the factors of 24 are written in ascending order.

So, the factors of 24 : 1,2,3,4,6,8,12,24.There are total 8 factors. The pair of factors that will multiply to 24: First and last factor will multiply together and result in original number. Similarly, second and second last factors will multiply to 24. And so on. So, there are total 4 pair factors of that multiply to 24. That is, total number of factor/2 will be the number of pairs of factors that multiply to form the given number.

Total number of factors of 2268 = (2+1) ×  (4+1) ×  (1+1) = 30 factors. 15 pairs of factors will multiply to 2268. Hence, 15 times the original number gets generated. So, the product of all factors is (2268)^{15 .}

3. Sum of reciprocals of all factors = Sum of all factors/Original number

The sum of factors of a number N=  a^p ×  b^q ×  c^r  

can be written as  (a^p+1 – 1)/(a – 1) × (b^q+1 – 1)/(b –1)  × (c^r+1 – 1)/(c  –1). Using this formula, the sum of factors comes out to be 6776. This result when divided by original number, that is 2268, will result in Sum of reciprocals of all the factors. So, Sum of reciprocals of all the factors is 6776/2268.
4. The total number of factors of a number N = a^p x b^q x c^r … = (p+1).(q+1).(r+1)…

Total number of factors of 2268 = (2+1) ×  (4+1) ×  (1+1) = 30 factors.

5. Total number of factor/2 will result in pairs of factor that multiply to form the original number. So, there are 15 pair of factors that multiply to 2268.

6. There are 2 factors that is 2⁰ and 2² that are perfect squares for 2. Similarly, there are 3 factors 3⁰ , 3² and 3⁴ that are perfect squares for 3. Finally, there is only 1 factor that is 7⁰ that is perfect square for the number 7. Hence, total combinations comes out to be 2 × 3 × 1 = 6 perfect square factors.

7. Odd factors will not include 2. So, for the number 2 there is only 1 factor 2⁰ will be taken into consideration. Next, for number 3 all the 5 factors 3⁰ , 3¹ , 3² , 3³ and 3^{4 .} Finally, for the number 7 all the two factors 7⁰ and 7^{1  .} The total combination comes out to be 1 × 5 × 2 = 10 factors that are odd.

8. Composite number of factors = Total number of factors – ( number of Prime factors + 1). Need to subtract 1 also from the total number of factors as 1 is neither prime nor composite. So Composite factors = 30 – ( 3 + 1) = 30 – 4 = 26 composite factors.

9. Even number of factors = Total number of factors – Odd number of factors. That is 30 – 10 = 20 even number of factors.

10. When the number is even, we need to divide the number by 4 because only even ×  even condition needs to be taken care of (click here to watch the video). So, the resultant comes out to be 567. Now, we just need to find the pair of factors that will multiply to result 567. That is total number of factors / 2. Total number of factors of 567 is 10. So, 10/2 = 5 ways of writing the number as a difference of two squares.