krit.club logo

Multiples and Factors - Factor Trees

Grade 5CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

•

A factor is a number that divides another number exactly without leaving a remainder. In a factor tree, we visualize this by breaking a large number into two smaller numbers that multiply to give the original number, representing them as branches extending downwards.

•

Prime Numbers are the building blocks of factor trees. A prime number has exactly two factors: 11 and itself (e.g., 2,3,5,7,112, 3, 5, 7, 11). In a factor tree diagram, once you reach a prime number at the end of a branch, you circle it to show that this branch has finished growing.

•

Composite Numbers are numbers that have more than two factors, such as 4,6,8,9,104, 6, 8, 9, 10. In a factor tree, these numbers are the ones that continue to sprout new branches until every path ends in a prime number.

•

The Factor Tree Method is a visual diagram used to find the prime factorization of a number. It looks like an upside-down tree where the main number is at the top (the root) and the prime factors are at the bottom (the leaves).

•

Branching involves splitting a composite number into any two of its factors. For example, to factorize 1212, you could draw two lines from 1212 pointing to 22 and 66, or to 33 and 44. The final result will always be the same regardless of which factors you start with.

•

Prime Factorization is the final step where you write the number as a product of all the prime numbers found at the ends of the branches. Visually, these are all the circled numbers at the tips of the 'leaves' of your factor tree.

•

The Uniqueness of Prime Factorization means that every composite number has exactly one set of prime factors. Even if different students start their trees with different factor pairs, they will always end up with the same group of circled prime numbers at the bottom.

📐Formulae

Number=Factor1×Factor2\text{Number} = \text{Factor}_1 \times \text{Factor}_2

Prime Factorization of 12=2×2×3=22×3\text{Prime Factorization of } 12 = 2 \times 2 \times 3 = 2^2 \times 3

Dividend=Divisor×Quotient+Remainder\text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder} (where Remainder = 00 for factors)

💡Examples

Problem 1:

Construct a factor tree for the number 3636 and write its prime factorization.

Solution:

Step 1: Start with 3636 at the top. Split it into two factors, for example, 6×66 \times 6. \nStep 2: Since 66 is a composite number, split both 66s into factors. 6=2×36 = 2 \times 3. \nStep 3: Now we have 2,3,2,32, 3, 2, 3. Since 22 and 33 are prime numbers, we circle them and stop. \nStep 4: Collect all the circled numbers: 2,3,2,32, 3, 2, 3. \nStep 5: Write as a product: 36=2×2×3×336 = 2 \times 2 \times 3 \times 3.

Explanation:

We used the factor tree method to break down the composite number 3636 into its simplest building blocks. Even if we started with 4×94 \times 9, the final prime factors would still be two 22s and two 33s.

Problem 2:

Find the prime factors of 4848 using the factor tree method.

Solution:

Step 1: Split 4848 into 22 and 2424. Circle 22 (it is prime). \nStep 2: Split 2424 into 22 and 1212. Circle 22. \nStep 3: Split 1212 into 22 and 66. Circle 22. \nStep 4: Split 66 into 22 and 33. Circle both 22 and 33 as they are prime. \nStep 5: The prime factorization is 2×2×2×2×32 \times 2 \times 2 \times 2 \times 3.

Explanation:

In this example, we consistently branched out using the smallest prime number 22 until only prime numbers remained. The product 2×2×2×2×32 \times 2 \times 2 \times 2 \times 3 equals 4848.