The number of odd divisors of 120 is 4.
Understanding Odd Divisors
To find the number of odd divisors of any integer, the most effective method involves prime factorization. An odd divisor is a number that is not divisible by 2. Therefore, when determining the odd divisors, we only consider the odd prime factors of the given number.
Step-by-Step Calculation for 120
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Prime Factorization:
First, express 120 as a product of its prime factors.
$120 = 2 \times 60$
$120 = 2 \times 2 \times 30$
$120 = 2 \times 2 \times 2 \times 15$
$120 = 2 \times 2 \times 2 \times 3 \times 5$
So, the prime factorization of 120 is $2^3 \times 3^1 \times 5^1$. -
Identify Odd Prime Factors:
From the prime factorization $2^3 \times 3^1 \times 5^1$, the odd prime factors are 3 and 5. The factor $2^3$ is ignored when counting odd divisors because any multiple of 2 would result in an even number. -
Calculate the Number of Odd Divisors:
To find the number of odd divisors, we consider the exponents of the odd prime factors. For each odd prime factor, add 1 to its exponent and then multiply these results.- For the prime factor 3, the exponent is 1. So, (1 + 1) = 2.
- For the prime factor 5, the exponent is 1. So, (1 + 1) = 2.
Number of odd divisors = (exponent of 3 + 1) $\times$ (exponent of 5 + 1)
Number of odd divisors = (1 + 1) $\times$ (1 + 1) = $2 \times 2 = 4$.
Listing the Odd Divisors of 120
The odd divisors are formed by taking combinations of powers of 3 (up to $3^1$) and 5 (up to $5^1$), along with $3^0$ and $5^0$ (which both equal 1).
| Factor from $3^0, 3^1$ | Factor from $5^0, 5^1$ | Odd Divisor |
|---|---|---|
| $3^0 = 1$ | $5^0 = 1$ | $1 \times 1 = 1$ |
| $3^0 = 1$ | $5^1 = 5$ | $1 \times 5 = 5$ |
| $3^1 = 3$ | $5^0 = 1$ | $3 \times 1 = 3$ |
| $3^1 = 3$ | $5^1 = 5$ | $3 \times 5 = 15$ |
As shown in the table, the odd divisors of 120 are 1, 3, 5, and 15. There are indeed 4 odd divisors. This method provides a clear and systematic way to determine the count of odd divisors for any given integer. For more information on prime factorization and divisors, you can refer to resources like Khan Academy's introduction to prime factorization or similar number theory tutorials.
