Math Problem Statement

Find the Highest Common Factor of x = 3 * 5^3 * 11^3 * 13^5 and y = 5 * 7 * 13^5. Give your answer in index form.

Solution

Let's find the Highest Common Factor (HCF) of the given expressions:

Expression for xx:

x=3×53×113×135x = 3 \times 5^3 \times 11^3 \times 13^5

Expression for yy:

y=5×7×135y = 5 \times 7 \times 13^5

Steps:

  1. Identify common prime factors:

    • xx and yy share the common prime factors 5 and 13.
  2. Take the lowest powers of the common factors:

    • For 55, the powers are 535^3 in xx and 515^1 in yy. The lowest power is 515^1.
    • For 1313, both have 13513^5. The lowest power is 13513^5.
  3. HCF in index form: HCF=51×135\text{HCF} = 5^1 \times 13^5

Thus, the HCF is 5×1355 \times 13^5.

Would you like further details or explanations on any part of this? Here are some additional questions you might consider:

  1. How do you find the HCF of algebraic expressions with multiple variables?
  2. What happens if there are no common factors between two expressions?
  3. Can the HCF include non-prime factors in certain cases?
  4. How does the HCF differ from the least common multiple (LCM)?
  5. How would you express the solution if xx and yy had more terms or variables?

Tip: Always remember to compare the lowest powers of each common factor when calculating the HCF of two expressions.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Number Theory
Prime Factorization
Highest Common Factor (HCF)
Indices

Formulas

HCF = product of lowest powers of common factors

Theorems

Fundamental Theorem of Arithmetic

Suitable Grade Level

Grades 8-10