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Number - Factors, Multiples, and Primes

Grade 6IB

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

🔑Concepts

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Factors are whole numbers that can be multiplied together to produce another number; they divide into the target number without leaving a remainder. Visually, factors can be organized into a 'Factor Rainbow' where pairs of factors (like 11 and 2424, 22 and 1212) are connected by colored arcs, showing they multiply to the same product.

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Multiples are the result of multiplying a number by an integer. For example, the multiples of 44 are 4,8,12,16,…4, 8, 12, 16, \dots. Visually, multiples can be represented on a number line as equal 'jumps' starting from zero, illustrating the concept of skip-counting.

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A Prime Number is a natural number greater than 11 that has exactly two factors: 11 and itself (e.g., 2,3,5,7,112, 3, 5, 7, 11). Visually, a prime number of objects can only be arranged in a single straight line (a 1×n1 \times n array) and cannot form any other rectangular shape.

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A Composite Number is a natural number greater than 11 that has more than two factors (e.g., 4,6,8,94, 6, 8, 9). Visually, composite numbers are 'rectangular numbers' because their total units can be rearranged into at least one rectangle other than a single row, such as 66 dots forming a 2×32 \times 3 grid.

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Prime Factorization is the process of expressing a composite number as a product of its prime factors. This is most clearly visualized using a 'Factor Tree,' where the number sits at the top and branches out into factor pairs until every branch ends in a prime 'leaf' that cannot be split further.

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The Highest Common Factor (HCF) or Greatest Common Factor (GCF) is the largest factor shared by two or more numbers. In a visual Venn Diagram, the HCF is found by multiplying all the prime factors that sit inside the overlapping intersection of the circles representing each number.

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The Least Common Multiple (LCM) is the smallest multiple that is common to two or more numbers. Visually, using a Venn Diagram of prime factors, the LCM is calculated by multiplying every single number visible across both circles, including the overlapping intersection.

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Square Numbers are the result of multiplying a whole number by itself (e.g., 3×3=93 \times 3 = 9). Visually, these numbers can be arranged into a perfect square grid, where the number of rows equals the number of columns.

📐Formulae

Product of two numbers=HCF(a,b)×LCM(a,b)\text{Product of two numbers} = HCF(a, b) \times LCM(a, b)

Prime Factorization Form: n=p1a×p2b×p3c…\text{Prime Factorization Form: } n = p_{1}^{a} \times p_{2}^{b} \times p_{3}^{c} \dots

Area of a Square (Square Number):A=s2\text{Area of a Square (Square Number)}: A = s^2

Divisibility by 3:∑digits÷3=integer\text{Divisibility by 3}: \sum \text{digits} \div 3 = \text{integer}

💡Examples

Problem 1:

Find the Prime Factorization of 120120 using index notation.

Solution:

Step 1: Divide by the smallest prime factor, 22: 120=2×60120 = 2 \times 60. Step 2: Divide 6060 by 22: 60=2×3060 = 2 \times 30. Step 3: Divide 3030 by 22: 30=2×1530 = 2 \times 15. Step 4: Divide 1515 by the next prime factor, 33: 15=3×515 = 3 \times 5. Step 5: 55 is a prime number, so we stop. Step 6: Write as a product: 2×2×2×3×52 \times 2 \times 2 \times 3 \times 5. Step 7: Convert to index notation: 23×3×52^3 \times 3 \times 5.

Explanation:

We use the factor tree method to break the number down into its prime building blocks. Index notation is used to simplify the repeated multiplication of the prime factor 22.

Problem 2:

Determine the HCF and LCM of 2424 and 3636.

Solution:

Step 1: Prime factorization of 2424: 24=23×324 = 2^3 \times 3. Step 2: Prime factorization of 3636: 36=22×3236 = 2^2 \times 3^2. Step 3: To find the HCF, take the lowest power of common prime factors: 22×31=4×3=122^2 \times 3^1 = 4 \times 3 = 12. Step 4: To find the LCM, take the highest power of all prime factors present: 23×32=8×9=722^3 \times 3^2 = 8 \times 9 = 72.

Explanation:

The HCF represents the largest shared divisor, found by taking the intersection of prime factors. The LCM represents the first shared multiple, found by taking the union of all prime factors at their highest frequency.