The number 90 has 6 odd factors.
Understanding the factors of a number, especially distinguishing between odd and even factors, is a fundamental concept in number theory. An odd factor is any factor that is not divisible by 2.
What Are Factors?
Factors of a number are integers that divide the number evenly, leaving no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. Identifying these factors helps in various mathematical computations and problem-solving scenarios.
Finding the Factors of 90
To determine the odd factors of 90, we first need to list all its factors. A systematic way to do this is through prime factorization.
Prime Factorization of 90
Prime factorization breaks down a number into its prime components.
The prime factorization of 90 is:
$$90 = 2 \times 45$$
$$90 = 2 \times 3 \times 15$$
$$90 = 2 \times 3 \times 3 \times 5$$
$$90 = 2^1 \times 3^2 \times 5^1$$
This factorization shows that 90 is composed of one '2', two '3's, and one '5'.
Listing All Factors of 90
Every factor of 90 is a product of these prime components, where the exponents are less than or equal to the exponents in the prime factorization.
| Combinations | Factor | Odd/Even |
|---|---|---|
| $2^0 \times 3^0 \times 5^0$ | $1 \times 1 \times 1 = 1$ | Odd |
| $2^0 \times 3^1 \times 5^0$ | $1 \times 3 \times 1 = 3$ | Odd |
| $2^0 \times 3^2 \times 5^0$ | $1 \times 9 \times 1 = 9$ | Odd |
| $2^0 \times 3^0 \times 5^1$ | $1 \times 1 \times 5 = 5$ | Odd |
| $2^0 \times 3^1 \times 5^1$ | $1 \times 3 \times 5 = 15$ | Odd |
| $2^0 \times 3^2 \times 5^1$ | $1 \times 9 \times 5 = 45$ | Odd |
| $2^1 \times 3^0 \times 5^0$ | $2 \times 1 \times 1 = 2$ | Even |
| $2^1 \times 3^1 \times 5^0$ | $2 \times 3 \times 1 = 6$ | Even |
| $2^1 \times 3^2 \times 5^0$ | $2 \times 9 \times 1 = 18$ | Even |
| $2^1 \times 3^0 \times 5^1$ | $2 \times 1 \times 5 = 10$ | Even |
| $2^1 \times 3^1 \times 5^1$ | $2 \times 3 \times 5 = 30$ | Even |
| $2^1 \times 3^2 \times 5^1$ | $2 \times 9 \times 5 = 90$ | Even |
The complete list of factors for 90 is: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90.
Identifying Odd Factors
An odd factor is a factor that is not divisible by 2. In terms of prime factorization, an odd factor must not include 2 as one of its prime components. This means we only consider the powers of the odd prime factors (3 and 5) from the prime factorization of 90 ($2^1 \times 3^2 \times 5^1$).
To find the number of odd factors, we consider only the exponents of the odd prime factors:
- For the prime factor 3, the exponent is 2. The possible powers are $3^0, 3^1, 3^2$ (3 options).
- For the prime factor 5, the exponent is 1. The possible powers are $5^0, 5^1$ (2 options).
The total number of odd factors is the product of the number of options for each odd prime factor's power:
$$ \text{Number of odd factors} = (2+1) \times (1+1) = 3 \times 2 = 6 $$
The odd factors of 90 are formed by taking combinations of these powers:
- $3^0 \times 5^0 = 1 \times 1 = 1$
- $3^1 \times 5^0 = 3 \times 1 = 3$
- $3^2 \times 5^0 = 9 \times 1 = 9$
- $3^0 \times 5^1 = 1 \times 5 = 5$
- $3^1 \times 5^1 = 3 \times 5 = 15$
- $3^2 \times 5^1 = 9 \times 5 = 45$
As evident from the list, the odd factors of 90 are 1, 3, 5, 9, 15, and 45. Counting these, we find there are indeed six odd factors. Through systematic analysis, it becomes clear that the number of odd factors of 90 is 6.
