Factors of 289 Using Prime Factorization
What Are the Factors of 289?
If the number 289 is divided by a natural number such that the remainder is zero, this natural number is known as a factor of 289. Here, the quotient obtained on division is also a factor of 289.
The factors of 289 are 1, 17, and 289 because all these numbers divide the number 289 evenly.
Deriving the Factor List of 289
Divisor | Is the number a factor of 289? | Multiplication equation |
|---|---|---|
1 | Yes, 1 is a factor of every number. | 1 \( \times \) 289 = 289 |
2 | No, 289 is not even. | - |
3 | No, 2 + 8 + 9 = 19, 19 is not divisible by 3. | - |
4 | No, 289 \( \div \) 4 = 72R1 | - |
5 | No, ones digit is not 0 or 5. | - |
6 | No, 289 is not even. | - |
7 | No, 289 \( \div \) 7 = 41R2 | - |
8 | No, 289 \( \div \) 8 = 35R9 | - |
9 | No, 289 \( \div \) 9 = 32R1 | - |
10 | No, 289 \( \div \) 10 = 28R9 | - |
11 | No, 289 \( \div \) 11 = 26R3 | - |
12 | No, 289 \( \div \) 12 = 24R1 | - |
13 | No, 289 \( \div \) 13 = 22R3 | - |
14 | No, 289 \( \div \) 14 = 20R9 | - |
15 | No, 289 \( \div \) 15 = 19R4 | - |
16 | No, 289 \( \div \) 16 = 18R1 | - |
17 | Yes, 289 \( \div \) 17 = 17R0 | 17 \( \times \) 17 = 289 |
The multiplication equation has started repeating the numbers that result in 289 as the product. We can stop checking the factor pairs after 17.
So, the factors of 289 are 1, 17, and 289.
[Note: If we divide 289 by 20, we get 14 and 9 as the quotient and the remainder, respectively. So, 20 is not a factor of 289. Similarly, you can check for other numbers as well.]
The Prime Factorization of 289
If a natural number can be expressed as a product of prime numbers, these prime numbers are known as the prime factors of that natural number. The process of writing this multiplication of prime factors is known as prime factorization.
The prime factors of 289 can be calculated by the factor tree method as shown below.
So, the prime factorization of 289 can be represented as, 289 = 17 \( \times \) 17 or 17\( ^2 \), and the only prime factor of 289 is 17.
Read More:
Factor Pairs of 289
A factor pair of a number is a set of any of its two factors such that their product is the number itself. A factor pair can be negative or positive.
The positive factor pairs of 289 are (1, 289) and (17, 17).
1 \( \times \) 289 = 289
17 \( \times \) 17 = 289
Positive Factor Pairs of 289
Positive Factors of 289 | Positive Factor Pairs of 289 |
|---|---|
1 \( \times \) 289 | (1, 289) |
17 \( \times \) 17 | (17, 17) |
Rapid Recall
Factors of 289 are 1, 17, and 289.
Solved Factors of 289 Examples
Example 1: Find the common factor(s) of 289 and 290.
Solution:
Factors of 289 are 1, 17, and 289
Factors of 290 are 1, 2, 5, 10, 29, 145, and 289
Therefore, the common factor of 289 and 290 is 1.
Example 2: Find the greatest common factor of 136 and 289.
Solution:
Factors of 136 = 1, 2, 4, 8, 17, 34, 68, and 136
Factors of 289 = 1, 17, and 289.
So, the common factors of 136 and 289 are 1 and 17.
Therefore, the greatest common factor of 136 and 289 is 17.
Example 3: Jenny was studying geometrical shapes and found a rectangle of area 289 square centimeters. What are the possible dimensions of this rectangle?
Solution:
The area of the rectangle is 289 cm\( ^2 \).
Area of rectangle = length \( \times \) width
289 = length \( \times \) width
So, the dimensions of the rectangle will be the factor pairs of 289, which are, (1, 289), (17, 17), and (289, 1)
Therefore, the possible dimensions of the rectangle are:
Length in centimeters | Width in centimeters |
|---|---|
1 | 289 |
17 | 17 |
289 | 1 |
The factors of 289 are the numbers that divide 289 evenly, with the remainder being 0. The number 289 is exactly divisible by 1, 17, and 289 only. So, the factors of 289 are 1, 17, and 289.
Composite numbers are those numbers which have a factor other than 1 and itself, that is, a composite number has more than two factors.
The division of any positive number by 1 results in zero as the remainder, and the quotient is the number itself. Hence, 1 is a factor of all positive numbers.
Yes, because we get 289 by multiplying 17 with 17. So, we can say that 289 is a perfect square number.
The factors of 289 are 1, 17, and 289. Here, 17 is the only prime number among them. so, 17 is a prime factor of 289.