Factors of 180 by Prime Factorization, Factor Tree & Division Method
Factors of 180 are the numbers that can be multiplied together to get the product \(180\). In other words, the factors of \(180\) are the numbers that divide \(180\) without leaving a remainder. The factors of \(180\) are \(1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90\), and \(180\).
What are Factors of 180?
There are a total of eighteen factors of 180 and they are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90 and 180. Here 1 is the smallest and 180 is the largest factor of 180. All these factors are categorized into prime factors, pair factors (both negative and positive) and common factors.
The factors of \(\textbf{180}\) are the numbers that can be multiplied together to get the product \(180\). In other words, the factors of \(180\) are the numbers that divide \(180\) without leaving a remainder. The factors of \(180\) are \(1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90\), and \(180\). These are the only positive integers that can be multiplied together in different ways to get the product \(180\).
To check why \(1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90\), and \(180\) are the factors of \(180\), we can perform a simple division. When we divide \(180\) by \(1\), we get \(180\) with no remainder. When we divide \(180\) by \(2\), we also get \(90\) with no remainder. And when we divide \(180\) by \(3\), we get \(60\) with no remainder. Similarly, check for other factors of \(180\) that, they are divisible by \(180\) without leaving any remainder.
Therefore, \(1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90\), and \(180\) are the only factors of \(180\).
Prime Factors of 180
Prime numbers in maths are all positive integers that can only be evenly divided by \(1\) and itself. Prime factors of \(180\) are all the prime numbers that when multiplied together equal \(180\). We know that \(180\) is not a prime number, but it can be expressed as the product of prime numbers.
The process of finding the prime factors of \(180\) is called prime factorization of \(180\). To get the prime factors of \(180\), you divide \(180\) by the smallest prime number possible. Then you take the result from that and divide that by the smallest prime number. Repeat this process until you end up with \(1\), as shown in the figure.
So, the prime factorization of \(180\) is \(2 \times 2 \times 3 \times 3 \times 5\).
Therefore, the prime factors of \(180\) are \(2, 3\), and \(5\).
Composite Factors of 180
Composite numbers can be defined as numbers that have more than two factors. Numbers that are not prime are composite numbers because they are divisible by more than two numbers.
We know that the factors of \(180\) are \(1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90\), and \(180\). Composite factors of \(180\) are \(4, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90\), and \(180\). A number can be classified as prime or composite depending on their divisibility.
The number \(180\) has an even number at its unit's place, therefore it is divisible by \(2\). So, we can say that \(180\) is a composite number and will surely have more than two factors. Similarly, we check for other factors of \(180\). Therefore, the composite factors of \(180\) are \(4, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90\), and \(180\).
Pair Factors of 180
Pair factors of a number are the pairs of two numbers that when multiplied together give the original number. \(180\) can be expressed as a product of two numbers in all possible ways. In each product, both multiplicands are the factors of \(180\).
The table below shows the factor pairs of \(180\):
Factors | Pair Factors |
\(1 \times 180 = 180\) | \((1, 180)\) |
\(2 \times 90 = 180\) | \((2, 90)\) |
\(3 \times 60 = 180\) | \((3, 60)\) |
\(4 \times 45 = 180\) | \((4, 45)\) |
\(5 \times 36 = 180\) | \((5, 36)\) |
\(6 \times 30 = 180\) | \((6, 30)\) |
\(9 \times 20 = 180\) | \((9, 20)\) |
\(10 \times 18 = 180\) | \((10, 18)\) |
\(12 \times 15 = 180\) | \((12, 15)\) |
Therefore, from the above table we see that - \((1, 180)\), \((2, 90)\), \((3, 60)\), \((4, 45)\), \((5, 36)\), \((6, 30)\), \((9, 20)\), \((10, 18)\), and \((12, 15)\) are the only pair factors of \(180\).
Similarly, we can find the negative factor pairs of \(180\) as follows:
Factors | Negative Factor Pairs |
\(-1 \times -180 = 180\) | \((-1, -180)\) |
\(-2 \times -90 = 180\) | \((-2, -90)\) |
\(-3 \times -60 = 180\) | \((-3, -60)\) |
\(-4 \times -45 = 180\) | \((-4, -45)\) |
\(-5 \times -36 = 180\) | \((-5, -36)\) |
\(-6 \times -30 = 180\) | \((-6, -30)\) |
\(-9 \times -20 = 180\) | \((-9, -20)\) |
\(-10 \times -18 = 180\) | \((-10, -18)\) |
\(-12 \times -15 = 180\) | \((-12, -15)\) |
Therefore, from the above table we see that negative factor pairs of \(180\) are \((-1, -180)\), \((-2, -90)\), \((-3, -60)\), \((-4, -45)\), \((-5, -36)\), \((-6, -30)\), \((-9, -20)\), \((-10, -18)\), and \((-12, -15)\).
Common Factors of 180
Common factors of two or more numbers are the numbers that divide both numbers leaving zero as the remainder. The common factors of \(180\) are the factors that \(180\) shares with another number. Let us understand this with the help of an example.
Example: Find the common factors of \(180\) and \(90\).
First write the factors of \(180\) and the factors of \(90\).
Factors of \(180\) = \(1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90\), and \(180\).
Factors of \(90\) = \(1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45\) and \(90\).
So, the common factors of \(180\) and \(90\) are \(1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45\) and \(90\).
