What is the smallest number by 288 must be multiplied so that the product is perfect square?

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What are the factors of 288 and 228?
Is 288 a perfect square?
What are the factors of 288 that sum up to 42.?
What are the negative factors of 288?

Did you know that 288 = 1+4+27+256 = 11 + 22 + 33 + 44 ? Like all even numbers (except for 2), 288 has more than one factor. However, it is one of those rare numbers, for which the exponents in its prime factorization are 5 and 2. In this lesson, we will calculate the factors of 288, prime factors of 288, and factors of 288 in pairs along with solved examples for a better understanding.

Factors of 288: 1, 2, 3, 4, 6, 8, 9, 12, 16,18, 24, 32, 36, 48, 72, 96,144, 288
Prime Factorization of 288: 2 × 2 × 2 × 2 × 2 × 3 × 3

The numbers that divide 288 completely, without a remainder, are its factors.

In other words, the factors of 288 are the numbers that get multiplied in pairs to create it. Look at the rainbow below to find which two numbers create 288 on multiplication.

We factor those numbers repeatedly to get all the factors of 288.

Explore factors using illustrations and interactive examples:

Factors of 96: The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 128: The factors of 128 are 1, 2, 4, 8, 16, 32, 64 and 128
Factors of 108: The factors of 108 are 1, 2, 3 ,4, 6, 9, 12, 18, 27, 36, 54 and 108.
Factors of 180: The factors of 180 are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180
Factors of 72: The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72

We know that 288 is a composite number and must be having many factors. Using the divisibility rules, we find the factors of 288 starting from 1.

288 ÷ 1=288
Check if 288 is divisible by 2, as it is an even number.
288 ÷ 2 =144
288 ÷ 3 = 96
288 ÷ 4 =72
288 is not divisible by 5
288 ÷ 6 = 48
288 is not divisible by 7
288 ÷ 8 = 36
288 ÷ 9 = 32
288 is not divisible by 10 and 11
288 ÷ 12 = 24
288 is not divisible by 13, 14 and 15
288 ÷ 16 = 18

All the divisors that we have got here, completely divide 288 without any remainder. Thus, all these divisors become the factors of 288.

"Prime factorization is to express a composite number as the product of its prime factors." To find the prime factors of a composite number, keep dividing it by the prime numbers one by one, until there are no composite factors left. The prime factorization of 288 is shown below by the ladder method.

288 is a composite number. We know that 288 is divisible by 2. 144 ÷ 2. Thus, we break it further into 2 factors. 72 ÷ 2
72 can be factored as 36 ÷ 2
36 is factored as 18 ÷ 2
18 is factored as 9 ÷ 2
9 is factored as 3 ÷ 3
Since we have no more composite number, we can stop here. Below is the factor tree of 288.

288 = 2 × 2 × 2 × 2 × 2 × 3 × 3= 25. × 32. The prime factors of 288 are 2 and 3. Now that we have done the prime factorization of our number, we can multiply them and get all the other possible factors.

Tips and Tricks

Use the ladder method or factor tree to find the prime factorization of a composite number.
Do the prime factorization of the number, multiply them and get all the possible composite factors.
288 = 2 × 2 × 2 × 2 × 2 × 3 × 3= 25 × 32. To determine the number of factors of 288, add one to each of these exponents and multiply them. (5+1) (2+1) = 6 × 3 = 18
Use the rainbow method to find the factor-pairs.

The pair of numbers which gives 288 when multiplied are known as factor pairs of 288. Following are the factors of 288 in pairs.

Positive Factor Pairs of 288	Negative Factor Pairs of 288
1 × 288	-1 x -288
2 × 144	-2 × -144
3 × 96	-3 × -96
4 × 72	-4 × -72
6 × 48	-6 × -48
8 × 36	-8 × -36
9 × 32	-9 × -32
12 × 24	-12 × -24
16 × 18	-16 × -18

288 is a composite number with 18 factors.
The factors of a number divide it completely without any remainder. 288 ÷ 3 =96, with remainder = 0
The factors of 288 that add up to 6 = 1 + 2 + 3 . Factors of 288 that add up to 42 are 18 + 24

Example 1: Jane is given a few number cards numbered from 10 to 15, She is asked to find the factors of 288 between 10 and 15, Can you help her?

Solution:

To know if a number is a factor, we will check all the numbers between 10 and 15 to know which numbers completely divide 288 without leaving any remainder.

288 ÷ 10 = 28, Remainder = 8

288 ÷11 = 26, Remainder = 2

288 ÷ 12 = 24, Remainder = 0

288 ÷ 13 = 22, Remainder = 2

288 ÷ 14 = 20, Remainder = 8

288 ÷ 15 = 19, Remainder = 3

This shows that only 12 is a factor of 288.
Example 2: Paul has 288 cookies. He needs to put 9 cookies in each box. How many boxes will he use?

Solution:

9 cookies × number of boxes = 288 cookies

9 × _____= 288 He needs to find the missing factor.
9 × 32 = 288

Therefore, he will use 32 boxes.

Example 3: Mia builds a rectangular base mat with 288 unit blocks. She has built a row of 16 blocks. How many such rows of blocks does she need to arrange to get the full rectangular base?

Solution:

16 blocks × number of rows = 288 blocks

16 × ______= 288
Finding the missing factor, we get 16 × 18 = 288
Thus she needs 18 more such rows of 16 to complete the rectangle.

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The factors of 288 get multiplied together to make 288. 1 × 288, 2 ×144, 3 × 96, 4 × 72, 6 × 48, 8 × 36, 9 × 32, 12 × 24, 16 ×18 are the pairs that make 288.

So the factors of 288 are 1, 2, 3, 4, 6, 8, 9, 12, 16,18, 24, 32, 36, 48, 72, 96,144, 288.

What are the factors of 288 and 228?

The factors of 288 are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 72, 96, 144 and 288
The factors of 228 are 1, 2, 3, 4, 6, 12, 19, 38, 57, 76, 114, 228.