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Factorize pairs of numbers, find the HCF, and test if it divides them.

Mathematics

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4 parts
a

HCF=4, Yes, it divides both numbers.\text{HCF} = 4, \text{ Yes, it divides both numbers.}

b

HCF=6, Yes, it divides both numbers.\text{HCF} = 6, \text{ Yes, it divides both numbers.}

c

HCF=10, Yes, it divides both numbers.\text{HCF} = 10, \text{ Yes, it divides both numbers.}

d

HCF=12, Yes, it divides both numbers.\text{HCF} = 12, \text{ Yes, it divides both numbers.}

Step-by-step solution

12 steps

Here's how to factorize the pairs of numbers, find the product of their common prime factors (HCF), and test if it divides the given numbers.

Part a· 3 steps

8 and 12

  1. Find the prime factors of 8 and 12.

    8=2×2×28 = 2 \times 2 \times 2 12=2×2×312 = 2 \times 2 \times 3

  2. Identify the common prime factors and calculate their product (HCF).

    The common prime factors are 2×2=42 \times 2 = 4.

  3. Test if the product (4) exactly divides 8 and 12.

    8÷4=28 \div 4 = 2 12÷4=312 \div 4 = 3

    Both divisions are exact.

    HCF=4, Yes, it divides both numbers.\boxed{\text{HCF} = 4, \text{ Yes, it divides both numbers.}}

Part b

HCF=6, Yes, it divides both numbers.\text{HCF} = 6, \text{ Yes, it divides both numbers.}

18 and 24

  1. Find the prime factors of 18 and 24.

    18=2×3×318 = 2 \times 3 \times 3 24=2×2×2×324 = 2 \times 2 \times 2 \times 3

  2. Identify the common prime factors and calculate their product (HCF).

    The common prime factors are 2×3=62 \times 3 = 6.

  3. Test if the product (6) exactly divides 18 and 24.

    18÷6=318 \div 6 = 3 24÷6=424 \div 6 = 4

    Both divisions are exact.

    HCF=6, Yes, it divides both numbers.\boxed{\text{HCF} = 6, \text{ Yes, it divides both numbers.}}

Part c

HCF=10, Yes, it divides both numbers.\text{HCF} = 10, \text{ Yes, it divides both numbers.}

20 and 30

  1. Find the prime factors of 20 and 30.

    20=2×2×520 = 2 \times 2 \times 5 30=2×3×530 = 2 \times 3 \times 5

  2. Identify the common prime factors and calculate their product (HCF).

    The common prime factors are 2×5=102 \times 5 = 10.

  3. Test if the product (10) exactly divides 20 and 30.

    20÷10=220 \div 10 = 2 30÷10=330 \div 10 = 3

    Both divisions are exact.

    HCF=10, Yes, it divides both numbers.\boxed{\text{HCF} = 10, \text{ Yes, it divides both numbers.}}

Part d

HCF=12, Yes, it divides both numbers.\text{HCF} = 12, \text{ Yes, it divides both numbers.}

36 and 48

  1. Find the prime factors of 36 and 48.

    36=2×2×3×336 = 2 \times 2 \times 3 \times 3 48=2×2×2×2×348 = 2 \times 2 \times 2 \times 2 \times 3

  2. Identify the common prime factors and calculate their product (HCF).

    The common prime factors are 2×2×3=122 \times 2 \times 3 = 12.

  3. Test if the product (12) exactly divides 36 and 48.

    36÷12=336 \div 12 = 3 48÷12=448 \div 12 = 4

    Both divisions are exact.

    HCF=12, Yes, it divides both numbers.\boxed{\text{HCF} = 12, \text{ Yes, it divides both numbers.}}
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