Math Problem Statement

When written as the product of their prime factors: A = 2^2 × 3^3, B = 2^3 × 3^5, C = 2^3 × 5^2 × 7. Find: (i) the value of the square root of A, (ii) the lowest common multiple of A, B, and C, (iii) the highest common factor of A, B, and C. 16. (a) Find the prime factors of 1224.

Solution

Let's break down the problem in the image.

You are given the following prime factorizations of A, B, and C:

  • A = 22×332^2 \times 3^3
  • B = 23×352^3 \times 3^5
  • C = 23×52×72^3 \times 5^2 \times 7

The tasks are:

  1. Find the value of the square root of A.
  2. Find the lowest common multiple (LCM) of A, B, and C.
  3. Find the highest common factor (HCF) of A, B, and C.

Step-by-step solutions:

(i) The value of the square root of A:

To find the square root of A, you will take the square root of each prime factor with its exponent:

  • A = 22×332^2 \times 3^3
  • A=22×33\sqrt{A} = \sqrt{2^2 \times 3^3}

The square root of a product is the product of the square roots of each factor. So: A=22×33=2×31.5=2×3×3=63\sqrt{A} = \sqrt{2^2} \times \sqrt{3^3} = 2 \times 3^{1.5} = 2 \times 3 \times \sqrt{3} = 6 \sqrt{3}

(ii) The lowest common multiple (LCM) of A, B, and C:

The LCM is found by taking the highest power of each prime factor that appears in any of the numbers.

  • For 22, the highest power is 232^3 (from B and C).
  • For 33, the highest power is 353^5 (from B).
  • For 55, the highest power is 525^2 (from C).
  • For 77, the highest power is 77 (from C).

So, the LCM is: LCM(A,B,C)=23×35×52×7\text{LCM}(A, B, C) = 2^3 \times 3^5 \times 5^2 \times 7

(iii) The highest common factor (HCF) of A, B, and C:

The HCF is found by taking the lowest power of each prime factor that appears in all of the numbers.

  • For 22, the lowest power is 222^2 (from A).
  • For 33, the lowest power is 333^3 (from A and B).
  • The primes 5 and 7 do not appear in all three numbers, so they are not part of the HCF.

So, the HCF is: HCF(A,B,C)=22×33\text{HCF}(A, B, C) = 2^2 \times 3^3

Final Answers:

  1. The square root of A is 636 \sqrt{3}.
  2. The LCM of A, B, and C is 23×35×52×72^3 \times 3^5 \times 5^2 \times 7.
  3. The HCF of A, B, and C is 22×332^2 \times 3^3.

Let me know if you need further clarification or details on any of the steps!


Related Questions:

  1. How do you calculate the square root of a number written in prime factorization form?
  2. What is the method to find the least common multiple (LCM) using prime factorization?
  3. How do you find the highest common factor (HCF) using prime factorization?
  4. Can you calculate the square root of any prime factorization?
  5. What would happen if the prime factors of A, B, and C were different?

Tip:

When finding the LCM or HCF using prime factorizations, always remember:

  • For LCM, take the highest power of each prime factor.
  • For HCF, take the lowest power of each common prime factor.

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Math Problem Analysis

Mathematical Concepts

Prime Factorization
Square Roots
LCM (Least Common Multiple)
HCF (Highest Common Factor)

Formulas

Square root of a number: √(a × b) = √a × √b
LCM: Take the highest powers of all primes in the factorization
HCF: Take the lowest powers of all common primes in the factorization

Theorems

Prime Factorization Theorem
LCM and HCF based on prime factors

Suitable Grade Level

Grades 7-9