Question

which terms could have a greatest common factor of $5m^2n^2$? select two options. $m^5n^5$; $5m^4n^3$; $10m^4n$; $15m^2n^2$; $24m^3n^4$

Explanation:

To determine which terms have a greatest common factor (GCF) of \(5m^2n^2\), we analyze the coefficients and the exponents of \(m\) and \(n\) in each term:

Step 1: Analyze the coefficient (the number part)

The GCF has a coefficient of 5. So, the terms must have a coefficient that is a multiple of 5 (or 5 itself). Let's check the coefficients:

• \(m^5n^5\): Coefficient is 1 (not a multiple of 5)
• \(5m^4n^3\): Coefficient is 5 (multiple of 5)
• \(10m^4n\): Coefficient is 10 (multiple of 5)
• \(15m^2n^2\): Coefficient is 15 (multiple of 5)
• \(24m^3n^4\): Coefficient is 24 (not a multiple of 5)

Step 2: Analyze the exponent of \(m\)

The GCF has \(m^2\), so the exponent of \(m\) in the term must be at least 2 (since GCF takes the smallest exponent). Let's check:

• \(5m^4n^3\): Exponent of \(m\) is 4 (≥2)
• \(10m^4n\): Exponent of \(m\) is 4 (≥2)
• \(15m^2n^2\): Exponent of \(m\) is 2 (≥2)

Step 3: Analyze the exponent of \(n\)

The GCF has \(n^2\), so the exponent of \(n\) in the term must be at least 2 (since GCF takes the smallest exponent). Let's check:

• \(5m^4n^3\): Exponent of \(n\) is 3 (≥2)
• \(10m^4n\): Exponent of \(n\) is 1 (<2, so discard)
• \(15m^2n^2\): Exponent of \(n\) is 2 (≥2)

Now, let's verify the GCF for the remaining candidates (\(5m^4n^3\) and \(15m^2n^2\)):

For \(5m^4n^3\) and \(15m^2n^2\):

• Coefficient GCF: GCF of 5 and 15 is 5.
• \(m\) exponent GCF: GCF of 4 and 2 is 2.
• \(n\) exponent GCF: GCF of 3 and 2 is 2.

So, the GCF is \(5m^2n^2\), which matches the required GCF.

Also, let's check \(5m^4n^3\) and \(15m^2n^2\) individually with the GCF:

• For \(5m^4n^3\): When we divide by \(5m^2n^2\), we get \(m^2n\) (which is an integer coefficient term, so valid).
• For \(15m^2n^2\): When we divide by \(5m^2n^2\), we get 3 (which is an integer coefficient term, so valid).

Now, let's check the other option \(5m^4n^3\) with \(m^5n^5\) (coefficient 1, so GCF coefficient would be 1, not 5, so discard). \(10m^4n\) has \(n^1\), so GCF for \(n\) would be 1, not 2, so discard. \(24m^3n^4\) has coefficient 24, GCF with 5 is 1, so discard.

So the two terms are \(5m^4n^3\) and \(15m^2n^2\).

Answer:

B. \(5m^4n^3\), D. \(15m^2n^2\)