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Answer :
To find the common factors of the expressions [tex]\(24 m^5\)[/tex] and [tex]\(48 m^{10}\)[/tex], let's break down the process:
1. Identify the coefficients and their greatest common divisor (GCD):
- The coefficients of the expressions are 24 and 48.
- The GCD of 24 and 48 is 24. This means any numerical factor common to both expressions must be a factor of 24.
2. Identify the powers of [tex]\(m\)[/tex] and determine the lowest power present:
- The powers of [tex]\(m\)[/tex] in the expressions are 5 for [tex]\(24 m^5\)[/tex] and 10 for [tex]\(48 m^{10}\)[/tex].
- The lowest common power that can be a factor is [tex]\(m^5\)[/tex].
3. Check each given expression to see if they are common factors:
- [tex]\(16 m^2\)[/tex]: 16 is not a factor of 24, so this cannot be a common factor.
- [tex]\(12 m^3\)[/tex]: 12 is a factor of 24 and [tex]\(m^3\)[/tex] is less than or equal to [tex]\(m^5\)[/tex], so this is a common factor.
- [tex]\(12 m^5\)[/tex]: 12 is a factor of 24 and [tex]\(m^5\)[/tex] matches the minimum power, so this is a common factor.
- [tex]\(16 m^5\)[/tex]: 16 is not a factor of 24, so this cannot be a common factor.
- [tex]\(24 m^5\)[/tex]: 24 is a factor of itself, and [tex]\(m^5\)[/tex] matches the minimum power, so this is a common factor.
- [tex]\(24 m^{10}\)[/tex]: Although the numerical coefficient 24 is a factor, the power of [tex]\(m\)[/tex] here exceeds the lowest, [tex]\(m^5\)[/tex]. We need the common expression to match [tex]\(m^5\)[/tex], therefore this is not a common factor.
After checking, the common factors from the list provided are:
- [tex]\(12 m^3\)[/tex]
- [tex]\(12 m^5\)[/tex]
- [tex]\(24 m^5\)[/tex]
These expressions can divide both [tex]\(24 m^5\)[/tex] and [tex]\(48 m^{10}\)[/tex] evenly with respect to both their coefficients and powers of [tex]\(m\)[/tex].
1. Identify the coefficients and their greatest common divisor (GCD):
- The coefficients of the expressions are 24 and 48.
- The GCD of 24 and 48 is 24. This means any numerical factor common to both expressions must be a factor of 24.
2. Identify the powers of [tex]\(m\)[/tex] and determine the lowest power present:
- The powers of [tex]\(m\)[/tex] in the expressions are 5 for [tex]\(24 m^5\)[/tex] and 10 for [tex]\(48 m^{10}\)[/tex].
- The lowest common power that can be a factor is [tex]\(m^5\)[/tex].
3. Check each given expression to see if they are common factors:
- [tex]\(16 m^2\)[/tex]: 16 is not a factor of 24, so this cannot be a common factor.
- [tex]\(12 m^3\)[/tex]: 12 is a factor of 24 and [tex]\(m^3\)[/tex] is less than or equal to [tex]\(m^5\)[/tex], so this is a common factor.
- [tex]\(12 m^5\)[/tex]: 12 is a factor of 24 and [tex]\(m^5\)[/tex] matches the minimum power, so this is a common factor.
- [tex]\(16 m^5\)[/tex]: 16 is not a factor of 24, so this cannot be a common factor.
- [tex]\(24 m^5\)[/tex]: 24 is a factor of itself, and [tex]\(m^5\)[/tex] matches the minimum power, so this is a common factor.
- [tex]\(24 m^{10}\)[/tex]: Although the numerical coefficient 24 is a factor, the power of [tex]\(m\)[/tex] here exceeds the lowest, [tex]\(m^5\)[/tex]. We need the common expression to match [tex]\(m^5\)[/tex], therefore this is not a common factor.
After checking, the common factors from the list provided are:
- [tex]\(12 m^3\)[/tex]
- [tex]\(12 m^5\)[/tex]
- [tex]\(24 m^5\)[/tex]
These expressions can divide both [tex]\(24 m^5\)[/tex] and [tex]\(48 m^{10}\)[/tex] evenly with respect to both their coefficients and powers of [tex]\(m\)[/tex].
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