Factor, in arithmetic, is a number or algebraic expression that divides every quantity or expression (of which it is a factor) uniformly, which means without any remainder. For example, 4 and 2 are factors of 8, when multiplied together, the result is 8.
Similarly, factors of 144 would be all the pairs of numbers, which, when multiplied together, give the result as 144. Now the question that arises is what the factors of 144 and how to find them are?
To start with, the factors of 144 are 1, 4, 2, 3, 6, 8, 16, 9, 12, 18, 24, 36, 48, 72 and 144. Now that the 'what are the factors of 144' is solved, the next step would be to figure out how to find these factors, and there are various methods to figure out a number's factors, such as:
Prime Factorization of 144 by division method is the first way to calculate a number's factors.
We'll start by dividing the number, i.e., 144 by the smallest integer, which is 2.
The process will start with, 144 ÷ 2 = 72
Then, 72 ÷ 2 = 36
36 ÷ 2 = 18
18 ÷ 2 = 9
Now that two can't be divided with nine further without getting a decimal number, we'll move onto the next integer, that is, 3.
Now, 9 ÷ 3 = 3
And, 3 ÷ 3 = 1
144 =
2 |144
2 |72
2 |36
2 |18
3 |9
3 |3
1
144 = 2 x 2 x 2 x 2 x 3 x 3
When the final outcome is 1 that is when we stop the division process as we can't go further than this. So, the prime factors of 144 are written in a format of 2 x 2 x 2 x 2 × 3 x 3 or 24 x 32, where 2 and 3 are the prime numbers.
Similarly, we can group two numbers which when multiplied together gives us the result as 144, which is also called as Pair Factors.
POSITIVE PAIRS | NEGATIVE PAIRS |
1 × 144 = 144 | -1 × -144 = 144 |
2 × 72 = 144 | -2 × -72 = 144 |
3 × 48 = 144 | -3 × -48 = 144 |
4 × 36 = 144 | -4 × -36 = 144 |
6 × 24 = 144 | -6 × -24 = 144 |
8 × 18 = 144 | -8 × -18 = 144 |
9 × 16 = 144 | -9 × -16 = 144 |
12 × 12 = 144 | -12 × -12 = 144 |
When calculated and counted, we concluded that the number 144 has 15 positive as well as 15 negative factors. Thus, there are 30 factors of 144 in total.
The exponential form is a compact way of representing a number. For example, instead of writing 2 × 2 × 2 × 2 = 16, we can always express it as 24.
Similarly, the prime factorization of 144 using exponents would be, 2 × 2 × 2 × 2 × 3 × 3 = 24 × 32.
144 is a composite number as well as a perfect square. 144 could also be called a dozen dozens as well as it is 1 x 12. Another fact that 144 is a perfect square as the square root of the number 144 is 12. Thus, the square root of 144 is an integer and 144 is a perfect square.
144 is a perfect square, yes, but the sum of its digits is also a perfect square, i.e., 9 (1+4+4), the product of its digits is also a perfect square, i.e.16, and its reverse is also a perfect square, i.e., 441.
Factor pairs: 144 = 1 x 144, 3 x 48, 2 x 72, 4 x 36, 8 x 18, 6 x 24, 9 x 16, 12 x 12.
Factors of 144: 1, 4, 2, 3, 6, 8, 16, 9, 12, 18, 24, 36, 48, 72, 144.
Prime factorization: 144 = 2 x 2 x 2 x 2 x 3 x 3.
Prime Factorization of 144 using exponents: 2 x 2 x 2 x 2 × 3 x 3 = 24 × 32
What are the Types of Factoring Methods?
Four types of factoring methods:
GCF, Greatest Common Factor – is the largest positive integer that divides uniformly into all numbers with zero as the remainder. E.g., 6 is the Greatest Common Factor of 18, 30 and 42
Difference in two squares - In arithmetic terms, the distinction of squares is a squared quantity subtracted from another squared quantity. E.g., the theorem of (a+b)(a-b)=a²-b²
The Sum or Difference in 2 cubes - The sum or the difference present between two cubes can be factored into a specific product. E.g., the theorem of sum; (a+b)( a^{2}−ab+b^{2}) = a^{3} + b^{3} , the theorem of difference; (a−b)(a^{2}+ab+b^{2}) = a^{3} − b^{3}
Trinomials - in the form of x^{2} + bx + c can regularly be factored because it is the product of two binomials. Understand that a binomial is really a two-term polynomial. E.g, the trinomial theorem; ax^{2} + bx + c.
What is an Easy Way to Find Factors of a Number?
The method of prime factorization by division method would be the easiest way out, to identify all the factors of a number. You just have to start dividing your number with the smallest integer, i.e., two and take the process further till it can't be divided by 2, then you have to start with next smallest, i.e., 3 and the process will go like this till you get 1 as the remainder. If the number doesn’t get divided by 2 in the first place, try dividing it with 3,or 4, and so on...
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