Standard Form of a Number and its Applications

Standard Form of a Number and its Applications:

Any composite number can be written as the products of its prime factors, this product is called Standard form of the number.
24 = 2 × 2 × 2 × 3 = 2³ × 3

Application of standard form of numbers:-
1. Using the standard form of a number find the sum and number of factors of a number.

a) How to find Sum of a factors of a number:
Suppose, we have to find the sum of factors and number of factors of 1240.
1240 = 2³ × 31 × 5.

The sum of the factors will be given by:
(2⁰ + 2¹ + 2² + 2³) × (5⁰ + 5¹) × (31⁰ × 31¹)
= (15) × (6) × (32) = 2880.

In this process, we have created same number of distinct bracket as many distinct prime factors of the number contained and then each bracket is filled with the sum of all the powers of the respective prime number starting from 0 to highest power of the prime number contained in the standard form. Thus, for 1240, we create 3 brackets- one each for 2, 5 and 31.

b) How to find Number of factors of a given number:
Standard form of 1240 is given by,
1240 = 2³ × 31 × 5.
The sum of the factors will be given by:
(2⁰ + 2¹ + 2² + 2³) × (5⁰ + 5¹) × (31⁰ + 31¹)
Here, we count number of terms in the expansion.
In above expansion 1240 contains number of terms = 4 × 2 × 2 = 16
Then the number of factors = 16
The moment you can realize that 1240 = 2³ × 31 × 5; the answer for the number of factors can be got by (3 + 1) × (1 + 1) × (1 + 1) = 16.

2. Sum and number of even and odd number of factors:
For 1240 = 1240 = 2³ × 31 × 5.

The sum of the factors will be given by:
(2⁰ + 2¹ + 2² + 2³) × (5⁰ + 5¹) × (31⁰ + 31¹)

Then sum of even number of factors is given by,
(2¹ + 2² + 2³) × (5⁰ + 5¹) × (31⁰ + 31¹)
= 14 × 6 × 32
= 2688

To find the sum of even number of factors, we just have eliminated 20 from the original answer.
Then number of even factors = 3 × (1 + 1) × (1 + 1) = 12; since, 2⁰ is already eliminated so, we are not adding 1 to the first bracket.
Then sum of odd number of factors is given by,
(2⁰) × (5⁰ + 5¹) × (31⁰ + 31¹)
= 6 × 32
= 192.
For finding the sum of odd number of factors, we just have eliminated (2¹ + 2² + 2³) from the original answer.
Then number of odd factors = 1 × (1 + 1) × (1 + 1) = 4.

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