krit.club logo

Playing with Numbers - Factors and Multiples

Grade 6CBSE

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

🔑Concepts

•

Factors and Multiples: A factor of a number is an exact divisor of that number. For example, the factors of 1212 are 1,2,3,4,6,1, 2, 3, 4, 6, and 1212. A multiple of a number is a number obtained by multiplying it by a natural number. Imagine a grid where you can arrange 1212 dots into rectangles like 1×121 \times 12, 2×62 \times 6, or 3×43 \times 4; these dimensions represent the factors.

•

Prime and Composite Numbers: Numbers having exactly two factors (1 and the number itself) are called Prime Numbers (e.g., 2,3,5,7,112, 3, 5, 7, 11). Numbers having more than two factors are called Composite Numbers (e.g., 4,6,8,94, 6, 8, 9). You can visualize this using the 'Sieve of Eratosthenes', a grid where you cross out multiples of primes to reveal the remaining prime numbers.

•

Divisibility Rules: These are shortcuts to determine if a number is divisible by another without full division. For example, a number is divisible by 33 if the sum of its digits is a multiple of 33. A number is divisible by 66 if it is divisible by both 22 (ends in an even digit) and 33. Divisibility by 1111 is checked by finding the difference between the sum of digits at odd places and even places; if the result is 00 or divisible by 1111, the number is divisible by 1111.

•

Highest Common Factor (HCF): The HCF of two or more given numbers is the highest (or greatest) of their common factors. It is also known as GCF (Greatest Common Factor). Visually, if you draw a Venn diagram representing the factors of two numbers, the HCF is the product of the numbers found in the overlapping intersection of the two circles.

•

Lowest Common Multiple (LCM): The LCM of two or more given numbers is the lowest (or smallest) of their common multiples. Think of two runners on a circular track starting at the same time but at different speeds; the LCM represents the time/distance at which they will both be back at the starting point simultaneously.

•

Prime Factorization: This is the process of expressing a composite number as a product of its prime factors. This is often visualized using a 'Factor Tree', where the number branches out into factors until all the 'leaves' at the bottom of the branches are prime numbers.

•

Perfect Numbers: A number for which the sum of all its factors (excluding the number itself) is equal to the number is called a perfect number. For example, 66 is a perfect number because its factors are 1,2,1, 2, and 33, and 1+2+3=61 + 2 + 3 = 6.

📐Formulae

Product of two numbers=HCF×LCM\text{Product of two numbers} = \text{HCF} \times \text{LCM}

HCF(a,b)=a×bLCM(a,b)\text{HCF}(a, b) = \frac{a \times b}{\text{LCM}(a, b)}

Divisibility by 11 condition: ∣(Sum of digits at odd places)−(Sum of digits at even places)∣=0 or a multiple of 11\text{Divisibility by 11 condition: } |(\text{Sum of digits at odd places}) - (\text{Sum of digits at even places})| = 0 \text{ or a multiple of } 11

💡Examples

Problem 1:

Find the HCF of 2424 and 3636 using the prime factorization method.

Solution:

Step 1: Find the prime factorization of 2424. 24=2×2×2×3=23×324 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3 Step 2: Find the prime factorization of 3636. 36=2×2×3×3=22×3236 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2 Step 3: Identify the common prime factors with the lowest power. The common factors are 22 and 33. The lowest power of 22 is 222^2 and the lowest power of 33 is 313^1. Step 4: Multiply these common factors. HCF=2×2×3=12\text{HCF} = 2 \times 2 \times 3 = 12

Explanation:

To find the HCF, we break each number down into its basic prime 'building blocks' and then select the highest quantity of blocks that both numbers have in common.

Problem 2:

Find the LCM of 12,16,12, 16, and 2424 using the common division method.

Solution:

Step 1: Arrange the numbers in a row, separated by commas. Step 2: Divide by the smallest prime number (22) that divides at least one of the numbers. 2∣12,16,242 | 12, 16, 24 2∣6,8,122 | 6, 8, 12 2∣3,4,62 | 3, 4, 6 2∣3,2,32 | 3, 2, 3 2∣3,1,32 | 3, 1, 3 3∣1,1,13 | 1, 1, 1 Step 3: Multiply all the divisors used. LCM=2×2×2×2×2×3=24×3\text{LCM} = 2 \times 2 \times 2 \times 2 \times 2 \times 3 = 2^4 \times 3 LCM=16×3=48\text{LCM} = 16 \times 3 = 48

Explanation:

The common division method involves dividing all numbers simultaneously by prime factors until all quotients become 1. The product of all divisors gives the LCM.