4. PRACTICE Iustify Conclusions The GCF of any two even numbers is always even. Determine whether the statement is true or false. If true, explain why. If false, give a reason.

Question

UpStudy AI · Accepted Answer

**Statement:** The GCF of any two even numbers is always even.

**Answer:** **True.**

**Explanation:**

- **Definition of Even Numbers:** An even number is any integer that is divisible by 2. So, if you have two even numbers, say $$ A $$ and $$ B $$, both can be expressed as:
  $$
  A = 2 \times m \quad \text{and} \quad B = 2 \times n
  $$
  where $$ m $$ and $$ n $$ are integers.

- **Greatest Common Factor (GCF):** The GCF of two numbers is the largest number that divides both of them without leaving a remainder.

- **Common Factor of Even Numbers:** Since both $$ A $$ and $$ B $$ are divisible by 2, the number 2 is a common factor.

- **Implication for GCF:** Because 2 is a common factor, and both numbers are multiples of 2, any common factor they share must also be a multiple of 2. This means that their GCF cannot be odd; it must be even.

- **Example:** 
  - Consider $$ A = 8 $$ and $$ B = 12 $$.
  - Factors of 8: 1, 2, 4, 8
  - Factors of 12: 1, 2, 3, 4, 6, 12
  - Common factors: 1, 2, 4
  - **GCF:** 4 (which is even)

- **Conclusion:** Since the GCF must be a multiple of 2, it is always even when both numbers are even.
The GCF of any two even numbers is always even because both numbers are divisible by 2, making their greatest common factor also divisible by 2.

4. PRACTICE Iustify Conclusions The GCF of any two even numbers is always even. Determine whether the statement is true or false. If true, explain why. If false, give a reason.

Real Tutor Solution

Answer

Solution

Extra Insights

Related Questions

Latest Arithmetic Questions