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Playing with Numbers - Prime Factorisation

Grade 6CBSE

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

🔑Concepts

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Prime Numbers: A prime number is a natural number greater than 11 that has exactly two factors: 11 and the number itself. For example, 2,3,5,7,112, 3, 5, 7, 11 are prime. Visually, a prime number of items can only be arranged in a single row or column, as they cannot form a rectangular grid of smaller dimensions.

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Composite Numbers: A composite number is a natural number greater than 11 that has more than two factors. These numbers can be visualized as being constructed by multiplying smaller prime numbers together, like building blocks.

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Prime Factorisation: This is the process of expressing a composite number as the product of its prime factors. For instance, the number 1212 can be written as 2×2×32 \times 2 \times 3. Every number has a unique prime factorisation.

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Factor Tree Method: This is a visual diagram where you split a number into two factors, forming branches. If a factor is composite, it is split further into its own factors. The process stops when every branch ends in a prime number. Visually, it looks like an upside-down tree where the bottom leaves are all primes.

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Division Method: This method involves dividing the given number by the smallest possible prime number that divides it exactly. The resulting quotient is then divided again by the smallest prime number. This is repeated in a vertical ladder-like structure until the final quotient is 11.

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Fundamental Property: Any composite number can be represented as a product of prime numbers in only one way, except for the order of the prime factors. For example, 30=2×3×530 = 2 \times 3 \times 5 is considered the same factorisation as 30=5×2×330 = 5 \times 2 \times 3.

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Co-prime Numbers: Two numbers are called co-prime if they have only 11 as their common factor. For example, 44 and 99 are co-prime because their only common factor is 11, even though neither 44 nor 99 are prime numbers individually.

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The Number 1: The number 11 is unique because it has only one factor (itself). Therefore, it is classified as neither a prime number nor a composite number.

📐Formulae

Product of two numbers=HCF×LCMProduct\ of\ two\ numbers = HCF \times LCM

Dividend=(Divisor×Quotient)+RemainderDividend = (Divisor \times Quotient) + Remainder

Number=p1×p2×p3×...×pnNumber = p_1 \times p_2 \times p_3 \times ... \times p_n (where each pp is a prime factor)

💡Examples

Problem 1:

Find the prime factorisation of 4848 using the Factor Tree Method.

Solution:

Step 1: Start with 4848. Branch it into 2×242 \times 24.\nStep 2: 22 is prime, so branch 2424 into 2×122 \times 12.\nStep 3: 22 is prime, so branch 1212 into 2×62 \times 6.\nStep 4: 22 is prime, so branch 66 into 2×32 \times 3.\nStep 5: Both 22 and 33 are prime. The leaves are 2,2,2,2,32, 2, 2, 2, 3.\nTherefore, 48=2×2×2×2×3=24×348 = 2 \times 2 \times 2 \times 2 \times 3 = 2^4 \times 3.

Explanation:

We break down the composite numbers at each level until every branch ends in a prime number. The product of these 'leaf' numbers gives the prime factorisation.

Problem 2:

Find the prime factorisation of 126126 using the Division Method.

Solution:

Step 1: Divide 126126 by the smallest prime 22: 126÷2=63126 \div 2 = 63.\nStep 2: 6363 is not divisible by 22. Try the next prime 33: 63÷3=2163 \div 3 = 21.\nStep 3: Divide 2121 by 33: 21÷3=721 \div 3 = 7.\nStep 4: 77 is a prime number, so divide by 77: 7÷7=17 \div 7 = 1.\nStep 5: The divisors are 2,3,3,72, 3, 3, 7.\nTherefore, 126=2×3×3×7=2×32×7126 = 2 \times 3 \times 3 \times 7 = 2 \times 3^2 \times 7.

Explanation:

This method uses successive division by prime numbers starting from the smallest (2,3,5,7...2, 3, 5, 7...) until the result reaches 11.