Factors of 49 Using Prime Factorization
Introduction
The integers that can divide 49 completely without a remainder are known as the factors of 49. The number 49 has three factors in total and they are 1, 7, and 49. The numbers that are multiplied in pairs to produce 49 are called the factor pairs of 49. We will learn about the various factors of 49 in this article, specifically the positive factor pairs of 49 as well as the prime factorization of 49. We will then solve a few examples.
Using Divisibility Rules and Division to find the Factors of 49
Therefore, the factors of 49 are 1, 7, and 49 and the factor pairs for 49 are (1, 49), (7 ,7), and (49 ,1).
The Prime Factors of a number
The image below is what is known as a factor tree, and this can be used to learn about the prime factorization of 49, as 49 is a composite number.
The prime factorization of 49 is \(7~\times~7\) or \(7^2\). This means that 7 is the factor of 49. We can also say that 7 is the prime factor of 49.
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The Positive Factor Pairs of 49
The factor pairs of 49 are made up of two factors multiplied together to produce the number 49. The following are some examples of such numbers:
Solved Examples
Example 1: Find the common factors of 49 and 21.
Solution:
The factors of 49 are 1, 7, and 49.
The factors of 21 are 1, 3, 7, and 21.
Hence, the common factors of 49 and 21 are 1 and 7.
Example 2: Find the common factor of 49 and 35.
Solution:
The factors of 49 are 1, 7, and 49.
The factors of 35 are 1, 5, 7, and 35.
Hence, the common factors of 49 and 35 are 1 and 7.
Example 3: Alex has 49 ribbons and Dave has 46. They both make a bow using two ribbons. Their sister Sarah asks for a ribbon. But Alex and Dave tell her that they will give her whatever is remaining after making the bows. Do you think Sarah will get a ribbon?
Solution:
Number of ribbons with Dave \(=~46\)
Number of bows made by Dave \(=~\frac{46}{2}~=~23\) R0
Number of ribbons with Alex \(=~49\)
Number of bows made by Alex \(=~\frac{49}{2}~=~24\) R1
When 46 is divided by 2, we get 23 with no remainder. So Dave can make 23 bows and will have no ribbons left with him. On the other hand, when 49 is divided by 2, we get 24 with 1 as a remainder. So after Alex makes 24 bows he would be left with 1 ribbon which he can give to Sarah.
Hence, Sarah gets 1 ribbon from Alex.
Example 4: Find the common factors of 49 and 56.
Solution:
The factors of 49 are 1, 7, and 49.
The factors of 56 are 1, 2, 4, 7, 8, 14, 28, and 56.
Hence, the common factors of 49 and 56 are 1 and 7.
Example 5: Find the common factors of 49 and 98.
Solution:
The factors of 49 are 1, 7, and 49.
The factors of 98 are 1, 2, 7, 14, 49, and 98.
Hence, the common factors of 49 and 98 are 1, 7, and 49.
1, 7 and 49 are the factors of 49.
\(7~\times~7\) or \(7^2\) is the prime factorization of 49.
(1, 49), (7, 7), and (49, 1) are the positive factor pairs of 49.
Yes, 7 is a factor of 49. As 7 divides 49 exactly and leaves the remainder 0, 7 is a factor of 49.